Benefits of time dilation / length contraction pairing?

[JesseM]

If I try putting this in my own words, could you tell me if I've got it right?

As I understand it, the time dilation formula takes as its input the time between two events in a frame where there’s no space between them, i.e. two events colocal in the unprimed frame. When you plug into it this time and the speed the primed frame is moving relative to the unprimed, the formula t’ = t * gamma tells you the time in the primed frame between the two events.

The length contraction formula takes as its input the distance between two events that are simultaneous in the unprimed frame. When you plug into it this distance and the speed the primed frame is moving relative to the unprimed frame, the formula L’ = L/gamma tells you the distance in the primed frame between one of these events *and a third event*, which third event is simultaneous in the primed frame with the first, and in the unprimed frame is colocal with the second.

Yes, these look right to me. But to make the physical meaning of the second one a little more clear, you might point out that if the first event occurs along the worldline of the left end of an object at rest in the unprimed frame, and the second event occurs along the worldline of the same object's right end (so if the distance between the events is L in the unprimed frame, this must be the length of the object in the unprimed frame), that means that the third event also occurs along the worldline of the object's right end, so since the first and third event are simultaneous in the primed frame, L' must be the length of the object in the primed frame. The idea is that the "length" of an object in any frame is defined as the distance between its two ends at a single instant in that frame.

More generally, I guess I’ve been trying to understand what exactly the asymetry is, and where it comes from: whether a physical difference between time and space, an accident of the kind of questions asked in textbooks, or of the way the idea is expressed, or if there’s something about our relationship to time and space that makes us want to ask a different question of each--that is, something that makes this pair of not-directly-analogous questions more useful to us than any other combination of the four possible questions illustrated in the diagram. Does that make any sense?

I guess I would say the usefulness of these two equation is that in physics it is typical to calculate dynamics by taking as "initial conditions" a spatial arrangement of objects at a single moment in time (along with the instantaneous velocities at that time), and then use the dynamical equations of physics to evolve that initial state forward through time. So, it's useful to know the set of spatial coordinates an object occupies at a single instant which is where the length contraction equation comes in handy, and it's useful to know how much a clock will have moved forward if you evolve the initial state forward by some particular amount of coordinate time, which is where the time dilation equation comes in handy. The "spatial analogue of time dilation" and the "temporal analogue of length contraction" equations would come in more handy if we took as our "initial state" a surface of constant x rather than a surface of constant t, and then evolved this state forward by increasing the x coordinate and looking at how things changed in successive surfaces of constant x. But this points to a real difference between how the laws of physics treat time and space; in a deterministic universe the laws of physics do allow you to determine what the physical state will be in later surfaces of constant t if you know the physical state at an earlier surface of constant t, but they don't allow you to determine the state of a surface of constant x if all you know is the state of some other surface of constant x. I guess this is also related to the fact that in SR the worldlines are timelike, so if we assume no worldline has a start or end (no worldlines starting or ending at singularities as in GR, and no particle worldlines ending because the particle annihilates with another particle as in quantum theory), then a given worldline will pierce every surface of constant t precisely once, while a worldline can have no intersections with a surface of constant x, or one pointlike intersection, or multiple pointlike intersections, or an infinite collection of points on that worldline can lie on a single surface of constant x.

 

[JesseM]

As someone new to SR, the use of primed and unprimed coordinates has been a source of some confusion for me. I think that’s partly because, to begin with, I didn’t know which details of a textbook explanation were going to turn out to be the significant ones, and which were accidental to a particular author’s presentation. At first, I got the impression that the convention was simply that primed frame = rest frame. But then I encountered examples such as: “Consider a rod at rest in frame S’ with one end at x’2 and the other end at x’2 [...]” (Tipler & Mosca: Physics for Scientists and Engineers, 5e, extended version, p. 1274). From this, I guessed that the desciding factor in whether to call a frame primed or unprimed was the direction it was moving, a primed frame being one that moves in the positive x direction with respect to the unprimed frame, hence the signs used in versions of the full Lorentz transformation such as this:

x' = gamma (x – ut)
t' = gamma (t – ux/c^2)

x = gamma (x' + ut')
t = gamma (t' + ux'/c^2)

I think that’s the traditional choice of signs, isn’t it? But I don’t know yet how rigid that convention is. The practice of using primed coordinates to represent a frame moving in the positive x direction agrees with their use in discussions of Euclidian rotation for a frame rotated in the positive (counterclockwise) direction. At least, that’s how Euclidian rotation was introduced in the first books where I met it. But in Spacetime Physics, Taylor and Wheeler do it the other way around, using primed coordinates for a negative (clockwise) rotation. I suppose, looking on the bright side, the advantage to having a variety of presentation methods is that it, eventually, you can see by comparing them which details are the physically significant ones, and which a matter of convention. But to begin with it’s a lot of information to take in.

There isn't really any absolute convention about primed and unprimed, they're just ways of differentiating two distinct frames, although most authors seem to follow the conventions that you said you were most used to above. But as long as you understand the physical relations of the two frames that's all that really matters. For example, if someone writes $$\Delta t' = \Delta t * \gamma$$ (the most common form I've seen), then you know $$\Delta t$$ must refer to two events which occur at the same position in the unprimed frame, like two readings on a clock at rest in the primed frame; if someone instead wrote $$\Delta t = \Delta t' * \gamma$$, then unless they just made a mistake, you'd know they intended to refer to the time intervals between two events which occur at the same position in the primed frame. Likewise, if someone wrote the following as the Lorentz transformation:

x = gamma (x' - vt')
t = gamma (t' - vx'/c^2)

Then although this is different from how they're usually written, you can infer that this is just a situation where it's assumed the origin of the unprimed frame is moving at velocity v along the x' axis of the primed frame.

 

[neopolitan]

Remember a while back I talked about an apparatus I had. I have it and it is at rest relative to me.

Associated with this apparatus are a length and a time measurement. I called these L and t.

I give these to my buddy, and he sets off on a carriage with a speed of v (in a direction that is convenient so that the length I measured as L is parallel to the direction of motion).

My buddy will, if he checks, find a length and time measurement of L and t.

But while my buddy in motion measures L and t, I will work out that, because he is motion, the length is contracted. I call that L', because I already have an L (unprimed is my frame so I making primed my buddy's frame). I will also work out that, because he is in motion, my buddy's clock will have slowed down. What reads on his clock will less than what I read on mine. If confuses you, and god knows it confuses me, because you have to step back a bit from the intial t I had. So, let's do it another way.

Say I have two sets of the apparatus. I keep one, and give the other to my buddy. I know they are identical. I ask him to measure it lengthwise, he gets L and I get L. But if I compare my length to his length (and I can do this with lasers and time measurements in my frame), I will find that he is "confused". His length is actually $$L'=L / \gamma$$. (And yes, I know if he does the same thing, he will find that I am "confused".)

Time is a little more complex to describe, but equivalent to using lasers and time measurements in my frame. Using a very high quality telescope, I keep track of my buddy's apparatus, most specifically the clock. I note down two times on his clock, $$t'_{o}$$ and $$t'_{i}$$ along with the times that I make them (my times, my frame, unprimed). I have to take into account how long it took each of those displayed times on his clock to get to me.

I will find that $$\Delta t' = \Delta t / \gamma$$ (<- this is my equation, this is not time dilation!)

Now I know that when $$\Delta t$$ has elapsed in my frame, $$\Delta t'$$ elapses in his frame. It is not just ticks on clocks, or the time interval between two events - his time dimension is affected. And it is affected in the same way as his spatial dimension is affected. So any speed in his frame will be calculated using contracted length divided by shortened time which will give you the same result as using unaffected length divided by unaffected time. Picking appropriate values of L and $$\Delta t$$:

$$L / \Delta t = c = L' / \Delta t'$$

Does that help?

cheers,

neopolitan

 

[Rasalhague]

Wow, thanks for your answer Jesse. That's given me lots to think about! In everyday life we're more used to regarding future time as being what's unpredictable, so it's curious to think of "the state of a surface of constant x" as being more fundamentally impossible to deduce "if all you know is the state of some other surface of constant x". But suppose that, in a deterministic universe, you knew everything about a surface of constant x to some arbitrary degree of precision. You'd know the positions of particles in that surface and be able to say something about other surfaces of constant x by examining the forces operating on the particles in your surface. Admittedly there'd be multiple possible states of the rest of space that could be responsible for the state of your surface of constant x, but then you could likewise have different histories that lead to the same state for some surface of constant t. So is it something about the future specifically, and its predictability, that makes all the difference? If that's even a meaningful question...

Meanwhile, less philosophically, just to check I've understood: is the case where a worldline has multiple pointlike intersections with a surface of constant x only possible for a particle for which there's no inertial frame in which the particle can be said to be at rest (i.e. its worldline isn't a straight line)? (The other two cases you mention--no intersections, one pointlike intersection--being possible for a particle which can be described as being at rest in some inertial frame.)

 

[JesseM]

But while my buddy in motion measures L and t, I will work out that, because he is motion, the length is contracted. I call that L', because I already have an L (unprimed is my frame so I making primed my buddy's frame).

You seem to be confused about what "unprimed is my frame so I am making primed my buddy's frame" means. The length of your buddy's apparatus is contracted when measured in your frame, his apparatus is not contracted in his own frame, so it's totally wrong to call the contracted length L' if you just said your frame is unprimed! If your frame is unprimed, then any variable that refers to how something appears in your frame--like the coordinate distance between either ends of an apparatus at single instant of time in your frame--must be unprimed, regardless of whether the physical object that you're measuring is at rest in your frame or not. Remember, physical objects aren't "in" one frame or another, different frames are just different ways of assigning coordinates to events associated with any object in the universe. And it's true that, as you say, "you already have an L" if you previously defined L to be the length of the same apparatus in your frame when it was at rest relative to you, but that just mean you need some different unprimed symbol to refer to the length of the apparatus in your frame once you've given it to your buddy and it's at rest relative to him, like $$L_{cbb}$$ where "cbb" stands for "carried by buddy".

Perhaps this confusion about what quantities should be primed and what quantities should be unprimed is related to your (so far unexplained) belief that there is something "inconsistent" about the way the standard time dilation and length contraction equations are written?

And even if I changed your statement above to "I will work out that, because he is motion, the length is contracted. I call that $$L_{cbb}$$, because I already have an L", your statement would still be too vague, for exactly the same reason as the statement in your last post was too vague (I offered several possible clarifications so you could pick which one you meant, or offer a different clarification). If $$L_{cbb}$$ refers to the length of the apparatus in your frame when it's being carried by your buddy, and $$L'_{cbb}$$ refers to the length your buddy measures the apparatus to be using his own ruler (which is equal to L, the length you measured the apparatus to be using your ruler before you gave it to your buddy, when it was still at rest relative to you), then these will be related by the equation $$L_{cbb} = L'_{cbb} / \gamma$$, which is just the length contraction equation with slightly different notation. If on the other hand what your buddy "measures" is a distance of L' between two events using his apparatus, then the distance you measure between the same two events will not necessarily be L'/gamma, in fact it could even end up being larger than L'. So you really need to be specific about precisely what is being measured like I keep asking.

I will also work out that, because he is in motion, my buddy's clock will have slowed down. What reads on his clock will less than what I read on mine.

It is meaningless to compare two compare two clock readings unless A) the clocks are located at the same position at the moment you do the comparison, or B) you have specified which frame's definition of simultaneity you're using. Do you disagree? If not, which one are you talking about here? If it's B, and if you're using your own frame's definition of simultaneity, and if the clocks initially read the same time at some earlier moment in your frame, then I agree that at the later moment his clock will read less than yours. But again, you really need to be way more specific or you'll end up using inconsistent definitions in different statements and end up with conclusions that don't make any sense, as seems to be true of your "t' = t/gamma has to be true in order that L/t=c and L'/t'=c" argument.

Say I have two sets of the apparatus. I keep one, and give the other to my buddy. I know they are identical. I ask him to measure it lengthwise, he gets L and I get L.

"It" is too vague since you have two sets, but I assume you mean "I ask him to measure the apparatus at rest relative to him, while I measure the apparatus at rest relative to me, his value $$L'_{cbb}$$ is exactly equal to my value L." Correct?

But if I compare my length to his length (and I can do this with lasers and time measurements in my frame), I will find that he is "confused". His length is actually $$L'=L / \gamma$$. (And yes, I know if he does the same thing, he will find that I am "confused".)

Time is a little more complex to describe, but equivalent to using lasers and time measurements in my frame. Using a very high quality telescope, I keep track of my buddy's apparatus, most specifically the clock. I note down two times on his clock, $$t'_{o}$$ and $$t'_{i}$$ along with the times that I make them (my times, my frame, unprimed). I have to take into account how long it took each of those displayed times on his clock to get to me.

And when you "take into account how long it took", you are using your frame's measurement of the distance that his clock was from yours when it read each of these two times, and assuming that the light from his clock travels at c in your frame, and subtracting distance/c from the time on your clock when you actually saw these readings, is that correct? For example, if when your clock reads 10 seconds you look through your telescope and see his clock reading 6 seconds, and at this moment you see his clock is next to a mark that's 2-light seconds away from you on your ruler, then you'd say his clock "really" read 6 seconds at the moment your clock read 8 seconds, correct?

If that is what you mean by "take into account"--and please actually tell me yes or no if it's what you meant--then note that this is exactly the same as asking what times on your clock were simultaneous with his clock reading $$t'_{o}$$ and $$t'_{i}$$, using your own frame's definition of simultaneity. So note that although you didn't really respond to my list of possible clarifications, it appears that your meaning is exactly identical to the first one I offered, which I'll put in bold (in the original comment I was using unprimed to refer to the buddy's frame and primed to refer your frame, but since you appear to want to reverse that convention by making times on your buddy's clock primed, I'll change the quote to reflect the idea that times in your frame are unprimed and times in your buddy's are primed):

And what does "readouts of time elapsed are reduced ... according to me" mean? Does it mean that if you consider a time interval from some time $$t_0$$ to another time $$t_1$$ in your frame, then for a clock at rest in his frame, the difference (his clock's readout at the event on the clock's worldline that occurs at time $$t_1$$ in your frame) - (his clock's readout at the event on the clock's worldline that occurs at time $$t_0$$ in your frame) is smaller than the difference $$t_1 - t_0$$ which represents the size of the time interval in your frame? In that case I would agree. Does it mean that if you consider two arbitrary events A and B (like two events on the worldline of a photon), with A happening at $$t_0$$ and B happening at $$t_1$$ in your frame, then for a clock at rest in his frame, the difference (his clock's readout at the event on the clock's worldline that occurs simultaneously with B in your frame) - (his clock's readout at the event on the clock's worldline that occurs simultaneously with A in your frame) is smaller than the time interval between A and B in your frame? If so, this is exactly equivalent to the first one, so I'd agree with this too. But does it mean that if you consider the same two events A and B, then for a clock at rest in his frame, the difference (his clock's readout at the event on the clock's worldline that occurs simultaneously with B in his frame) - (his clock's readout at the event on the clock's worldline that occurs simultaneously with A in his frame) is smaller than the time interval between A and B in your frame? If so this is not true in general. And if the two events are events on the path of a photon that occur on either end of a measuring-rod of length L in his frame, then it is that last difference in his clock's readouts that you divide L by to get c.

I will find that $$\Delta t' = \Delta t / \gamma$$ (<- this is my equation, this is not time dilation!)

So, if according to your frame's definition of simultaneity, your clock's reading $$t_ 0$$ is simultaneous with your buddy's clock reading some time $$t'_0$$, and according to your frame's definition of simultaneity your clock's reading $$t_1$$ is simultaneous with your buddy's clock reading some time $$t'_1$$, and if $$\Delta t' = t'_1 - t'_0$$ and $$\Delta t = t_1 - t_0$$, then we get the equation $$\Delta t' = \Delta t / \gamma$$. Is that what you mean? If so, then yes, I agree, and as I said this is exactly equivalent to the statement from my earlier post that I bolded above. But in this case you are simply confused if you think this is any different from the standard time dilation equation--it only looks different because you've reversed the meaning of primed and unprimed from the usual convention and then divided both sides by gamma! Normally, if we want to take two events on the worldline of a clock (in this case your buddy's) and then figure out the time interval between these events in a frame where the clock is moving (in this case yours--of course, figuring out the time interval between these events in your frame is exactly equivalent to figuring out which readings on your clock these two events are simultaneous with in your frame and then finding the difference between the two readings on your clock), the usual convention is to call the first frame unprimed and the second frame primed, in which case we get the time dilation equation $$\Delta t' = \Delta t * \gamma$$. You have simply adopted the opposite convention, calling the first frame primed and the second frame unprimed, so the time dilation equation would just have to be rewritten as $$\Delta t = \Delta t' * \gamma$$ using this convention. And of course, if we now divide both sides by gamma, we get back the equation you offered, $$\Delta t' = \Delta t / \gamma$$. You can see that this is just a trivial reshuffling of the usual time dilation equation, not anything novel.

Now I know that when $$\Delta t$$ has elapsed in my frame, $$\Delta t'$$ elapses in his frame.

No you don't, not for an arbitrary pair of events! Say you pick two events A and B which don't occur on the worldline of his clock (they may be two events on the worldline of a light beam for example), but such that according to his frame's definition of simultaneity, A is simultaneous with $$t'_0$$ and B is simultaneous with $$t'_1$$. Then would you agree that the time interval between these events in his frame is $$\Delta t' = t'_1 - t'_0$$? And we also know that the time interval in your frame between the event of his clock reading $$t'_1$$ and the event of his clock reading $$t'_0$$ is related to this by $$\Delta t = \Delta t' * \gamma$$. But that doesn't mean the time interval in your frame between A and B is $$\Delta t = \Delta t' * \gamma$$! This is because although it's true that his frame's definition of simultaneity says that A is simultaneous with his clock reading $$t'_0$$ and B is simultaneous with his clock reading $$t'_1$$, your frame uses a different definition of simultaneity, so according to your frame's definition of simultaneity A may not be simultaneous with his clock reading $$t'_0$$ and B may not be simultaneous with his clock reading $$t'_1$$, so knowing the time-interval in your frame between his clock reading $$t'_0$$ and his clock reading $$t'_1$$ tells us nothing about the time interval in your frame between A and B.

Do you understand and agree with all this? Please tell me yes or no.

It is not just ticks on clocks, or the time interval between two events - his time dimension is affected. And it is affected in the same way as his spatial dimension is affected. So any speed in his frame will be calculated using contracted length divided by shortened time which will give you the same result as using unaffected length divided by unaffected time. Picking appropriate values of L and $$\Delta t$$:

$$L / \Delta t = c = L' / \Delta t'$$

Nope, you still are unable or unwilling to define what you are actually supposed to be measuring the length of and time-intervals between, "appropriate values" is hopelessly vague. Do L and L' represent the distance between a single pair of events on the worldline of a photon, as measured in each frame? Or are you measuring two separate photons with two separate apparatuses, so L is the distance between one pair of events as measured in your frame and L' is the distance between another pair as measured in your buddy's frame? Or is it something else entirely? And how about $$\Delta t$$ and $$\Delta t'$$, are you going with the definition I suggested earlier where $$\Delta t'$$ is the difference between two clock readings $$t'_1$$ and $$t'_0$$ on your buddy's clock, and $$\Delta t$$ is the difference between two clock readings $$t_1$$ and $$t_0$$ on your clock, where you have picked the readings so that according to your frame's definition of simultaneity $$t_1$$ is simultaneous with $$t'_1$$ and $$t_0$$ is simultaneous with $$t'_0$$? If not, can you be specific about what events you are taking "deltas" between? And if so, are any of these events on the clocks' worldlines supposed to be simultaneous with events on the worldline of a photon in some frame?

 

[JesseM]

Wow, thanks for your answer Jesse. That's given me lots to think about! In everyday life we're more used to regarding future time as being what's unpredictable, so it's curious to think of "the state of a surface of constant x" as being more fundamentally impossible to deduce "if all you know is the state of some other surface of constant x". But suppose that, in a deterministic universe, you knew everything about a surface of constant x to some arbitrary degree of precision. You'd know the positions of particles in that surface and be able to say something about other surfaces of constant x by examining the forces operating on the particles in your surface.

That's an interesting point, but consider the fact that it's quite possible to have a surface of constant x such that of the particles that cross it (and many particles may never cross a particular surface of constant x at all), each one crosses it only at a single point in spacetime. In this case you'd only know the instantaneous velocity of each particle at its crossing point, but I don't see how this would allow you to deduce the force unless you also knew the instantaneous acceleration (and keep in mind that in deterministic theories like classical electromagnetism, merely knowing the position and instantaneous velocity of each particle in a surface of constant t, along with the direction and magnitude of force field vectors in space in that surface, is sufficient to allow you to predict what will happen at later times, you don't need to know the instantaneous accelerations). Also consider that in principle it would be possible to have a surface of constant x where no particles crossed it at any point, even though particles did exist in that universe--I suppose you could still be told the direction and magnitude of force field vectors in this otherwise empty surface, since force fields like the electromagnetic field are imagined to fill all of space, but I don't think this would allow you to deduce the complete history of every particle in the universe (it's possible I could be wrong about this since I haven't actually seen any discussions of this question, though).

Meanwhile, less philosophically, just to check I've understood: is the case where a worldline has multiple pointlike intersections with a surface of constant x only possible for a particle for which there's no inertial frame in which the particle can be said to be at rest (i.e. its worldline isn't a straight line)? (The other two cases you mention--no intersections, one pointlike intersection--being possible for a particle which can be described as being at rest in some inertial frame.)

That's right, assuming of course that we're talking about the x-coordinate of an inertial reference frame.

 

[neopolitan]

"Picking appropriate values of $$L$$ and $$\Delta t$$" was too vague. The rest of what you were saying was akin to "You can't park four tanks on the rubber dingy you're designing".

Here's what I mean about picking appropriate values, pick any value of $$L$$, any value you like - in the real world you probably want a really big value, but this is hypothetical world, so it is not so important.

Then pick the value of $$\Delta t$$ so that L/$$\Delta t = c$$. If you haven't picked a really big value of $$L$$, then $$/Delta t$$ will be pretty damn small so that it will be challenging to take two readings $$t_{o}$$ and $$t_{i}$$ where $$t_{i} - t_{o} = \Delta t$$ - but we are in hypothetical world.

We have no argument about length contraction. But do you deny that when I use my readings from my buddy's clock, and take into account the motion that I know he has, that I will get a $$/Delta t'$$ which is shorter than mine?

Do you deny that the extent to which it is shorter is the same as the extent to which L' is shorter than L (where these are given by standard length contraction)?

I've described the events, they aren't simultaneous (and in fact, I don't care about simultaneity, I know the time readings on my buddy's clock are not simultaneous with the time readings on mine, the only thing I bother with, or need to bother with, is the extra time the second reading takes to get to me because he has moved during the time). Any discussion of simultaneity in this scenario is a distraction. If you must have some simultaneity, then try thinking that my seeing the time on my buddy's clock is simultaneous with my reading of the time on my clock, but even that is not necessary since I could use a splitframe camera and look at the results afterwards.

The bottom line, from you Jesse, is "there is no other way to do it" when the question is "what is the benefit with time dilation". It seems you truly think there is no other option. You have a very long winded way to say it, but I don't think there is any other way to interpret your approach to the original question. And yes, I haven't forgotten the original question.

cheers,

neopolitan

 

[JesseM]

Here's what I mean about picking appropriate values, pick any value of $$L$$, any value you like - in the real world you probably want a really big value, but this is hypothetical world, so it is not so important.

Then pick the value of $$\Delta t$$ so that L/$$\Delta t = c$$. If you haven't picked a really big value of $$L$$, then $$\Delta t$$ will be pretty damn small so that it will be challenging to take two readings $$t_{o}$$ and $$t_{i}$$ where $$t_{i} - t_{o} = \Delta t$$ - but we are in hypothetical world.

Are you just picking a value of $$\Delta t$$ out of thin air, with no connection to anything physical (so you could just as easily pick a $$\Delta t$$ such that L/$$\Delta t$$ = 5c or any number you wish), or is it supposed to represent the time interval between some specific pair of events, like $$t_{o}$$ representing the time a photon passes next to one end of an object of length L which is at rest in your frame, and $$t_{i}$$ representing the time that photon passes next to the other end of the same object?

We have no argument about length contraction. But do you deny that when I use my readings from my buddy's clock, and take into account the motion that I know he has, that I will get a $$/Delta t'$$ which is shorter than mine?

What does "use my readings from my buddy's clock, and take into account the motion that I know he has" mean? This is something I specifically asked you about in my last response to you (post 75):

And when you "take into account how long it took", you are using your frame's measurement of the distance that his clock was from yours when it read each of these two times, and assuming that the light from his clock travels at c in your frame, and subtracting distance/c from the time on your clock when you actually saw these readings, is that correct? For example, if when your clock reads 10 seconds you look through your telescope and see his clock reading 6 seconds, and at this moment you see his clock is next to a mark that's 2-light seconds away from you on your ruler, then you'd say his clock "really" read 6 seconds at the moment your clock read 8 seconds, correct?

If that is what you mean by "take into account"--and please actually tell me yes or no if it's what you meant--then note that this is exactly the same as asking what times on your clock were simultaneous with his clock reading $$t'_{o}$$ and $$t'_{i}$$, using your own frame's definition of simultaneity.

Can you please answer this question? And if the answer is "no, that's not what I meant" could you please tell me exactly what you do mean, preferably giving some sort of numerical example? For example, suppose your buddy's clock is moving at 0.6c in your frame and is initially right next to your, and when they are next to each other both your clock and his read 0 seconds; then later when your clock reads 16 seconds, you look through your telescope and see his clock reading 8 seconds, and you also see that when his clock reads 8 seconds it's next to a mark on your ruler that's 6 light-seconds away from you (so in your frame the event must have 'really' happened 6 seconds before you saw it through your telescope). How would the phrase "use my readings from my buddy's clock, and take into account the motion that I know he has" apply to this specific example. What would $$\Delta t'$$ be, and what would $$\Delta t$$ be?

I've described the events

Have you? Where? Are the events just the two readings on your buddy's clock?

they aren't simultaneous (and in fact, I don't care about simultaneity, I know the time readings on my buddy's clock are not simultaneous with the time readings on mine, the only thing I bother with, or need to bother with, is the extra time the second reading takes to get to me because he has moved during the time). Any discussion of simultaneity in this scenario is a distraction.

But if by "taking into account" the light delay, you mean taking the time on your clock (t=16 seconds in my example above) when you saw a reading on your buddy's clock (t'=8 seconds in the example) and then subtracting the distance/c that your buddy's clock was from you in your frame when it showed that reading (6 light-seconds/c = 6 seconds in the example) to get an earlier time on your clock (t=16-6=10 seconds), then this is physically equivalent (meaning you'll get the same answer for what the two clock readings would be) to just asking for the time on your clock that was simultaneous in your frame with the reading you saw on your buddy's clock (i.e. in the example, the reading of t'=8 seconds on your buddy's clock is simultaneous in your frame with the reading of t=10 seconds on your clock). If this is indeed what you meant by "taking into account", then do you agree that this is physically equivalent to my statement about simultaneity? Please answer yes or no.

The bottom line, from you Jesse, is "there is no other way to do it" when the question is "what is the benefit with time dilation".

No it isn't, because I don't even know what the physical meaning of the "it" that you want to do actually is, your posts are just too unclear for me to judge them right or wrong. It will help if you give clear yes-or-no answers to questions about what you're saying when I ask for them.

 

[neopolitan]

JesseM,

You fragment too much. It leads, inexorably, to loss of context. That's why I am not responding to your fragmenting.

Look back in previous posts and I explained what I meant about taking readings on my buddy's clock. I made mention of a telescope.

But you must have overlooked it in your apparent excitement to demolish any discussion (and I mean that, "discussion", not argument because to demolish an argument you have to make an effort to understand).

You make an effort to understand, pose an unfragmented question which indicates that you have made the slightest effort to understand, rather than attack, and I will answer it.

cheers,

neopolitan

 

[Rasalhague]

it's also possible to come up with a "temporal analogue for the length contraction equation" which looks like the length contraction equation but with $$\Delta t$$ substituted for L (this is more difficult to state in words, but it's basically the time-interval in the primed frame between two surfaces of constant t in the unprimed frame which have a temporal distance of $$\Delta t$$ in the unprimed frame, which is analogous to how length contraction tells you the spatial distance in the primed frame between two worldlines of constant position in the unprimed frame which have a spatial separation of L in the unprimed frame).

Suppose Alice and Bob are each wearing a watch. Bob, moving in the positive x direction in Alice's rest frame, at 0.6c, passes Alice and they synchronise watches. Some time later, Alice looks at her watch and wonders, "What time does Bob's watch say at the moment which in Bob's rest frame is simultaneous with me asking this question?" The answer is given by $$t_{B} = \gamma t_{A}$$, the standard time dilation formula. If Alice's watch says 4, Bob's will say 5. "How about that," thinks Alice. "To Bob, for whom my watch is moving, it's running slow."

Now suppose that Alice wonders, "What time does Bob's watch say at the moment which in my rest frame is simultaneous with me asking this question?" The answer to this is given by $$t_{B} = \frac{t_{A}}{\gamma}$$. If Alice's watch says 5, Bob's will say 4. "Fancy," thinks Alice. "From my perspective, Bob's watch, which is moving relative to me, is running slow."

And, of course, Bob can ask the equivalent questions about the time on Alice's watch with identical results by virtue of the fact that the two frames don't agree on which events are simultaneous (except for those that happen in the same place, such as their synchronisation).

But isn't Alice's second question none other than this exotic "temporal analogue for the length contraction equation"? She wants to know "the time-interval in the primed frame" (the time shown by Bob's watch, indicating a time interval along Bob's worldline) "between two surfaces of constant t in the unprimed frame" (one being the one which Alice and Bob's worldlines intersected when they synchronised watches, the other being Alice's present when she looks at her watch) "which have a temporal distance of $$\Delta t$$ in the unprimed frame" (the time shown by Alice's watch when she looks at it and makes her query).

Is Alice's second question in any way less natural than the first, or a less useful thing to ask of time than of space? I'm puzzled as to how it can be, if it is, as Jesse said, "just a trivial reshuffling of the usual time dilation equation"?

 

[JesseM]

You make an effort to understand, pose an unfragmented question which indicates that you have made the slightest effort to understand, rather than attack, and I will answer it.

I have made an effort to understand, and in fact the questions above are pretty clearly requests to nail down the meaning of your statements by asking if they match up with the precise definitions that I have suggested, for example:

And when you "take into account how long it took", you are using your frame's measurement of the distance that his clock was from yours when it read each of these two times, and assuming that the light from his clock travels at c in your frame, and subtracting distance/c from the time on your clock when you actually saw these readings, is that correct? For example, if when your clock reads 10 seconds you look through your telescope and see his clock reading 6 seconds, and at this moment you see his clock is next to a mark that's 2-light seconds away from you on your ruler, then you'd say his clock "really" read 6 seconds at the moment your clock read 8 seconds, correct?

If that is what you mean by "take into account"--and please actually tell me yes or no if it's what you meant--then note that this is exactly the same as asking what times on your clock were simultaneous with his clock reading $$t'_{o}$$ and $$t'_{i}$$, using your own frame's definition of simultaneity.

Can you please answer this question? And if the answer is "no, that's not what I meant" could you please tell me exactly what you do mean, preferably giving some sort of numerical example? For example, suppose your buddy's clock is moving at 0.6c in your frame and is initially right next to your, and when they are next to each other both your clock and his read 0 seconds; then later when your clock reads 16 seconds, you look through your telescope and see his clock reading 8 seconds, and you also see that when his clock reads 8 seconds it's next to a mark on your ruler that's 6 light-seconds away from you (so in your frame the event must have 'really' happened 6 seconds before you saw it through your telescope). How would the phrase "use my readings from my buddy's clock, and take into account the motion that I know he has" apply to this specific example. What would $$\Delta t'$$ be, and what would $$\Delta t$$ be?

If you see this line of questioning as simply an "attack" rather than an attempt to actually understand in precise terms the meaning of your phrase "take into account how long it took" (and thereby to figure out the precise physical relationship between the two intervals $$\Delta t$$ and $$\Delta t'$$ which appear in your equation L/$$\Delta t$$ = c = L'/$$\Delta t'$$), then I suppose that means you are simply too mistrusting of my motives to ever be interested in the process of actual communication with me (and 'communication' necessarily requires a willingness to clarify what the other person doesn't understand, especially in a discussion of physics where precise definitions are needed), in which case I take it there is basically nothing I could do other than nodding my head and agreeing with all your statements (even when I don't really understand what they mean) that would make you want to continue the discussion.

 

[JesseM]

Suppose Alice and Bob are each wearing a watch. Bob, moving in the positive x direction in Alice's rest frame, at 0.6c, passes Alice and they synchronise watches. Some time later, Alice looks at her watch and wonders, "What time does Bob's watch say at the moment which in Bob's rest frame is simultaneous with me asking this question?" The answer is given by $$t_{B} = \gamma t_{A}$$, the standard time dilation formula. If Alice's watch says 4, Bob's will say 5. "How about that," thinks Alice. "To Bob, for whom my watch is moving, it's running slow."

Now suppose that Alice wonders, "What time does Bob's watch say at the moment which in my rest frame is simultaneous with me asking this question?" The answer to this is given by $$t_{B} = \frac{t_{A}}{\gamma}$$. If Alice's watch says 5, Bob's will say 4. "Fancy," thinks Alice. "From my perspective, Bob's watch, which is moving relative to me, is running slow."

Here you can use the time dilation formula too. If the time dilation formula is written $$\Delta t' = \Delta t * \gamma$$, then that' formula is comparing the amount of time that's elapsed on a clock (whose rest frame is labeled the unprimed frame) with the amount of time that's elapsed in a frame where the clock is moving, with "time elapsed" in that frame being based on that frame's definition of simultaneity (and with this second frame being labeled primed). So in your second example, the clock is Bob's and the second frame whose definition of simultaneity you're using is Alice's, so you can just treat Bob's frame as the unprimed frame in the standard time dilation equation and Alice's question will be the same as asking for the time elapsed in the primed frame, meaning you're just substituting $$t_A$$ and $$t_B$$ into the time dilation equation giving $$t_A = t_B * \gamma$$. Of course, if you wish to divide both sides by gamma, you can get back the formula $$t_{B} = \frac{t_{A}}{\gamma}$$ you wrote above, but this is just a reshuffling of the time dilation equation.

But you do raise a great point which I hadn't thought of before, which is that the "temporal analogue of the length contraction equation" can always be used to calculate things that you'd normally use the time dilation equation to calculate, provided you reverse the meaning of which frame is primed and which frame is unprimed. Let me give a numerical example similar to yours. Suppose Bob is moving away from Alice at 0.6c and that both their clocks read 0 when they crossed paths as you suggested. But instead of starting Bob's time interval when his clock reads 0 as in your example, suppose we were interested in the time interval on Bob's clock that started with the event of his clock reading $$t_{B1}$$ = 8 seconds, and ended with his clock reading $$t_{B2}$$ = 12 seconds, so the length of the interval in Bob's frame is $$\Delta t_B$$ = ($$t_{B2}$$ - $$t_{B1}$$) = 4 seconds. If Alice wanted to know the time interval $$\Delta t_A$$ between these same two events in her frame, which is equivalent to wanting to know the time interval between the event $$t_{A1}$$ on her clock which is simultaneous in her frame with $$t_{B1}$$ (in this case $$t_{A1}$$ = 10 seconds) and the event $$t_{A2}$$ on her clock which is simultaneous in her frame with $$t_{B2}$$ (in this case $$t_{A2}$$ = 15 seconds), then she would plug these two different time intervals into the time dilation equation $$\Delta t' = \Delta t * \gamma$$, treating Bob's frame as unprimed and her frame as primed, which gives $$\Delta t_A = \Delta t_B * \gamma$$. If she wanted to reverse this and figure out the time interval $$\Delta t_B$$ on Bob's clock between two events on $$t_{B1}$$ and $$t_{B2}$$ on his clock's worldline that are simultaneous in her frame with two events on her clock's worldline $$t_{A1}$$ and $$t_{A2}$$ that are the beginning and end of a time interval $$t_A$$ (in the example above she would start with times 10 seconds and 15 seconds on her clock and then try to figure out how much time had elapsed on Bob's clock between these moments in her frame), she'd just divide the time dilation equation by gamma so it gives $$\Delta t_B$$ as a function of $$\Delta t_A$$, i.e. $$\Delta t_B = \frac{\Delta t_A}{\gamma}$$.

On the other hand, the "temporal analogue of length contraction" $$\Delta t' = \Delta t / \gamma$$ would be telling her something conceptually different, assuming she continues to treat Bob's frame as unprimed and her frame as primed. Basically, it would be saying "if you use $$t_{A1}$$ to label the time on Alice's clock that's simultaneous in Bob's frame with Bob's clock reading $$t_{B1}$$, and you use $$t_{A2}$$ to label the time on Alice's clock that's simultaneous in Bob's frame with Bob's clock reading $$t_{B2}$$, then the time interval on Alice's clock ($$t_{A2} - t_{A1}$$) is related to the time interval on Bob's clock ($$t_{B2} - t_{B1}$$) by the formula $$(t_{A2} - t_{A1}) = (t_{B2} - t_{B1}) / \gamma$$. If we use the same numbers for $$t_{B1}$$ and $$t_{B2}$$ on Bob's clock as before, namely $$t_{B1}$$ = 8 seconds and $$t_{B2}$$ = 12 seconds, then in this case we'd have $$t_{A1}$$ = 8*0.8 = 6.4 seconds (I just multiplied 8 by 0.8 because I know both clocks read 0 when they were next to each other and Alice's clock is moving at 0.6c in Bob's frame, so the standard time dilation equation tells me her clock is slowed by a factor of 0.8 in his frame) and $$t_{A2}$$ = 12*0.8 = 9.6 seconds. So the equation $$(t_{A2} - t_{A1}) = (t_{B2} - t_{B1}) / \gamma$$ does work here, since $$(t_{A2} - t_{A1}$$ = 9.6 - 6.4 = 3.2, $$t_{B2} - t_{B1}$$ is still 4 seconds, and gamma is still 0.8. But you can see that the time interval in Alice's frame we're talking about now (3.2 seconds) is different than the time interval in Alice's frame we were talking about when we were using the usual time dilation equation (5 seconds). But, that's only because we were treating Alice's frame as the primed frame in both equations! If we reverse the labels and treat Bob's frame as primed and Alice's frame as unprimed, then the standard time dilation equation $$\Delta t' = \Delta t * \gamma$$ does tell you that when 3.2 seconds have elapsed on Alice's clock, 4 seconds of time have passed in Bob's frame (or equivalently, if you look at the readings on Bob's clock that are simultaneous in Bob's frame with the two readings on Alice's clock, the difference between these two readings on Bob's clock is 4 seconds).

So I guess if you take the time dilation equation and divide both sides by gamma to solve for the interval in the primed frame, this is really just equivalent to taking the "temporal equivalent of length contraction" equation and reversing which frame we call primed and which we call unprimed. To me there's still a little bit of a conceptual difference though, in the sense that normally I think of these equations as relating a clock time-interval to a coordinate time-interval, with unprimed normally being the clock time-interval. For instance, when I read the time dilation equation $$\Delta t' = \Delta t * \gamma$$, I find it most natural to think that $$\Delta t$$ represents the difference between two clock-readings on a clock at rest in the unprimed frame, and then $$\Delta t'$$ represents the difference between the coordinate times of these two readings in the primed frame. Of course, because a clock at rest in the primed frame will keep time with coordinate time in that frame, this is equivalent to imagining there's also a clock at rest in the primed frame, and saying $$\Delta t'$$ represents the difference between two readings on the primed clock that are simultaneous in the primed frame with the two readings on the unprimed clock that were mentioned earlier. The first way of stating it just makes the usefulness of the time dilation equation more intuitive to me; as I said before, physics is all about setting up a spacetime coordinate system and then using equations to figure out how the state of objects in space changes as the time-coordinate increases.

 

[neopolitan]

JesseM responding to Rasalhague:

But you do raise a great point which I hadn't thought of before, which is that the "temporal analogue of the length contraction equation" can always be used to calculate things that you'd normally use the time dilation equation to calculate, provided you reverse the meaning of which frame is primed and which frame is unprimed.

<snip>

The first way of stating it just makes the usefulness of the time dilation equation more intuitive to me; as I said before, physics is all about setting up a spacetime coordinate system and then using equations to figure out how the state of objects in space changes as the time-coordinate increases.

Thanks, I think you've given an answer my original question. I think you have said this:

There is another way of approaching the relativistic effects other than time dilation-length contraction. That is to use "temporal analogue of the length contraction equation"-length contraction. However, using time dilation is more intuitive to you - and possibly also for the majority of people. That said, there is nothing inherently wrong with using a "temporal analogue of the length contraction equation" (although one must note that a different prime convention is required).​

Rasalhague has shown me that instead of:

What exactly is the greater utility of time dilation and length contraction equations which prevents the use of two contraction equations ...?

I should have asked:

What exactly is the greater utility of length contraction and time dilation equations which prevents the use of a length contraction equation and a temporal equivalent of the length contraction equation ...?

For that I thank you Rasalhague.

cheers,

neopolitan

 

[JesseM]

Thanks, I think you've given an answer my original question. I think you have said this:

There is another way of approaching the relativistic effects other than time dilation-length contraction. That is to use "temporal analogue of the length contraction equation"-length contraction. However, using time dilation is more intuitive to you - and possibly also for the majority of people. That said, there is nothing inherently wrong with using a "temporal analogue of the length contraction equation" (although one must note that a different prime convention is required).​

Yes, although I hadn't actually realized that the "temporal equivalent of length contraction" equation could be used to answer exactly the same physical questions as the time dilation equation until Rasalhague asked that. And it was the specificity of the way he asked the question that made me realize this--he was asking about particular events on the worldlines of two physical clocks and stating in which frame an event on one clock's worldline was supposed to be simultaneous with a corresponding event on the other clock's worldline. You say that you "should have asked" this question:

What exactly is the greater utility of length contraction and time dilation equations which prevents the use of a length contraction equation and a temporal equivalent of the length contraction equation ...?

But such a broadly-worded question would almost certainly not have led me to the same realization. This illustrates why I keep asking you to answer specifics about what you are saying, and I don't really understand why you are unwilling to grant these requests--is it that you don't like my attitude, is it that your ideas are mostly intuitive so you're not sure what the answers should be yourself, or something else? I really think a willingness to delve into specifics could allow us to make progress on things like the meaning of "L/$$\Delta t$$ = c = L'/$$\Delta t'$$" which I have been unable to make sense of so far, just as the specifics of Rasalhague's question allowed for progress on the issue of the uses of the "temporal analogue of length contraction" equation, so I hope that even if you decide you are not interested in continuing this line of discussion for whatever reason, you will at least consider that there may be a lesson here about the value of engaging with nitty gritty details when talking physics.

 

[neopolitan]

We were probably arguing at cross purposes, frustratingly enough for both of us. I thought you had taken that "temporal equivalent of length contraction" thing onboard a long time ago (it is in your diagram after all). So I was totally confused as to where you were coming from.

Since I thought you had understood the point and were still arguing it, it felt as if you were just trying to play games. That may have been a form of "tranference" (psychological term, relating to ascribing apparent motives of one person to another), since in real life I had a rather difficult person at work doing what I thought you were doing - playing dominance games through irrational argument.

I take your point about specifics. You may see that I have tried to be specific with figures in another thread.

May I ask why you had not come to the understanding that you just came to, when it seems that both Rasalhague and I did? This is not a hidden "you must be stupid" insult. I find you annoying, as you surely find me, but I don't find you stupid. What I am trying to do is see if you can identify, from the vantage point of someone who has only just came to this understanding, what prevents people from coming to this understanding naturally. Is there a block of some kind? If so, is it pedagogical or psychological?

(Clarification follows: I am distinguishing here between pedagogical and psychological, with a definition of "pedagogical" relating to how subjects are taught and "psychological" relating to the different ways in which people think and learn. Specific examples: "whole language" is a pedagogical method for teaching kids to read, moving away from phonics and instead recognition of whole words. As for "psychological", I am a visual, pattern identifying person which means that having a graph in front of me is more useful than a page of numbers. My visual, pattern identifying nature may lead me to link together all things that look the same (like all things with primes against them get grouped).)

This is the sort of discussion I really wanted back when I started the thread. Perhaps you might understand why I found the 80 or so posts in between frustrating, even if they were my own fault.

cheers,

neopolitan

 

[JesseM]

May I ask why you had not come to the understanding that you just came to, when it seems that both Rasalhague and I did? This is not a hidden "you must be stupid" insult. I find you annoying, as you surely find me, but I don't find you stupid. What I am trying to do is see if you can identify, from the vantage point of someone who has only just came to this understanding, what prevents people from coming to this understanding naturally. Is there a block of some kind? If so, is it pedagogical or psychological?

Sure, it basically comes from the way I had drawn it in that diagram I gave you, which was the first time I had even thought about the issue of a "temporal analogue for length contraction" (let's call it the TAFLC equation for short). Note that if we write the standard time dilation equation as $$\Delta t' = \Delta t * \gamma$$, I have no problem with the idea that you can divide both sides by gamma to get $$\Delta t = \Delta t' / \gamma$$ (call this the 'reversed time dilation equation'), which I think of conceptually as telling us the time elapsed on a clock at rest in the unprimed frame between two events on its worldline which we know are separated by a time-interval of $$\Delta t'$$ in the primed frame (I said basically the same thing about reshuffling the time dilation equation in post #61, the paragraph beginning with "Also"). But although this "reversed time dilation equation" looks exactly like the TAFLC equation $$\Delta t' = \Delta t / \gamma$$ except for the switch between primed and unprimed, I was mistakenly thinking that the physical meaning of $$\Delta t'$$ and $$\Delta t$$ in the TAFLC equation was totally different from either of the terms in the reversed time dilation equation. Again, the reason was how it was drawn in my diagram--I was thinking that $$\Delta t'$$ represented some weird notion of the temporal distance in the primed frame between two surfaces of simultaneity from the unprimed frame that crossed through readings on the worldline of the clock at rest in the unprimed frame which have a separation of $$\Delta t$$. Superficially the notion of taking the temporal distance in the primed frame between two surfaces of constant t in the unprimed frame seems pretty weird and disconnected from anything physical (at least it did to me), an idea created only because it was analogous with taking the spatial distance in the primed frame between two worldlines of constant x in the unprimed frame, which is what length contraction is about.

What I had failed to realize was that if we imagine a physical clock at rest in the primed frame, then the "temporal distance between surfaces of constant t from the unprimed frame" just represents the difference $$\Delta t'$$ between the clock's readings at the two points where its worldline intersects these surfaces of constant t from the unprimed frame, and that if we then shift our perspective back to the unprimed frame, $$\Delta t$$ is now just the coordinate time between two readings on the primed clock, so now this is exactly like how I'd conceptualize the physical meaning of the terms in the reversed time dilation equation except with the roles of primed and unprimed reversed. So, this is one or two mental steps from what the TAFLC seemed to mean based on my diagram, and I didn't see the connection until I started working through a numerical example in response to Rasalhague's question. Also, it didn't help that I was used to conceptualizing the standard and reversed time dilation equations as relating a clock time-interval on a clock at rest in the unprimed frame with a coordinate time-interval in the primed frame, rather than normally thinking in terms of a clock at rest in the primed frame too. I was aware intellectually of the fact that the coordinate time in the primed frame between two events A and B could be rephrased in terms of readings on a physical clock at rest in the primed frame, specifically the difference between the reading that was simultaneous with A and the reading that was simultaneous with B according to the prime frame's definition of simultaneity. But that seemed like a more complicated way of thinking about the physical meaning of $$\Delta t'$$ (you can see it took me longer to write it out) so I usually just thought of it in terms of coordinate time.

 

[neopolitan]

Sure, it basically comes from the way I had drawn it in that diagram I gave you, which was the first time I had even thought about the issue of a "temporal analogue for length contraction" (let's call it the TAFLC equation for short). Note that if we write the standard time dilation equation as $$\Delta t' = \Delta t * \gamma$$, I have no problem with the idea that you can divide both sides by gamma to get $$\Delta t = \Delta t' / \gamma$$ (call this the 'reversed time dilation equation'), which I think of conceptually as telling us the time elapsed on a clock at rest in the unprimed frame between two events on its worldline which we know are separated by a time-interval of $$\Delta t'$$ in the primed frame (I said basically the same thing about reshuffling the time dilation equation in post #61, the paragraph beginning with "Also"). But although this "reversed time dilation equation" looks exactly like the TAFLC equation $$\Delta t' = \Delta t / \gamma$$ except for the switch between primed and unprimed, I was mistakenly thinking that the physical meaning of $$\Delta t'$$ and $$\Delta t$$ in the TAFLC equation was totally different from either of the terms in the reversed time dilation equation. Again, the reason was how it was drawn in my diagram--I was thinking that $$\Delta t'$$ represented some weird notion of the temporal distance in the primed frame between two surfaces of simultaneity from the unprimed frame that crossed through readings on the worldline of the clock at rest in the unprimed frame which have a separation of $$\Delta t$$. Superficially the notion of taking the temporal distance in the primed frame between two surfaces of constant t in the unprimed frame seems pretty weird and disconnected from anything physical (at least it did to me), an idea created only because it was analogous with taking the spatial distance in the primed frame between two worldlines of constant x in the unprimed frame, which is what length contraction is about.

What I had failed to realize was that if we imagine a physical clock at rest in the primed frame, then the "temporal distance between surfaces of constant t from the unprimed frame" just represents the difference $$\Delta t'$$ between the clock's readings at the two points where its worldline intersects these surfaces of constant t from the unprimed frame, and that if we then shift our perspective back to the unprimed frame, $$\Delta t$$ is now just the coordinate time between two readings on the primed clock, so now this is exactly like how I'd conceptualize the physical meaning of the terms in the reversed time dilation equation except with the roles of primed and unprimed reversed. So, this is one or two mental steps from what the TAFLC seemed to mean based on my diagram, and I didn't see the connection until I started working through a numerical example in response to Rasalhague's question. Also, it didn't help that I was used to conceptualizing the standard and reversed time dilation equations as relating a clock time-interval on a clock at rest in the unprimed frame with a coordinate time-interval in the primed frame, rather than normally thinking in terms of a clock at rest in the primed frame too. I was aware intellectually of the fact that the coordinate time in the primed frame between two events A and B could be rephrased in terms of readings on a physical clock at rest in the primed frame, specifically the difference between the reading that was simultaneous with A and the reading that was simultaneous with B according to the prime frame's definition of simultaneity. But that seemed like a more complicated way of thinking about the physical meaning of $$\Delta t'$$ (you can see it took me longer to write it out) so I usually just thought of it in terms of coordinate time.

Thanks for that. With some things going on the background it took me some time to digest.

There is something which I find curious. It is a criticism of the pedagogy not of you nor of what time dilation is actually representing.

Note that you are "used to conceptualizing the standard and reversed time dilation equations as relating a clock time-interval on a clock at rest in the unprimed frame with a coordinate time-interval in the primed frame". It's quite a complex thing to internalise. When being taught, or trying to teach oneself, it is going to be a real uphill struggle to grasp that particular nature of the standard time dilation equation.

I certainly struggled with it and it was not helped that I have "back to fundamentals" sort of approach to mathematics which I applied to SR by reading a translation of Einstein's 1905 paper (I use the one at http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf" ). I noted, and agonised over the fact that one of the standard equations is shown in mathematical form (length contraction) but the other is only given in words (on page 10) and this directly follows what was to me the far more intuitive equation - a form of "TAFLC" with $$\tau$$ instead of t'.

I do think there is a source of confusion there. I'd be willing to accept that it just me, but it seems there are many people with some problem or another with SR, which seems really odd. Why SR? Some are kooks for sure, but there are many people who seem to be otherwise able to maintain perfectly normal lives apart from an intuitive feeling that something is just not quite right about SR.

I won't say his name, and sadly he has probably passed away with cancer by now, but a professor in a city not far from where I lived up until recently was the lead lecturer for relativity at his university. He generously gave me many hours of his time to discuss my concerns and proposed solutions, and admitted that really, he didn't fully understand it. He did not stop me or explain that my concerns about time dilation were invalid, because he had never intuitively grasped the principles the standard way either. There is a fellow in southern Europe, another physics professor, albeit in a different field who expressed stronger views than I during our discussions that there was something amiss with SR. (I don't think SR is wrong but I do think it could be taught better.) A quantum physics professor in southern England also felt I was onto something with my arguments.

If professors of physics don't grasp SR properly, what chance do the average visitors to these forums have?

To make sure I am not presenting a biased account, I should clarify that at least four professors I corresponded with gave clear indications that they grasped SR well enough as taught (at least enough as to not be intrigued by my concerns), but sadly they had no time to go into it in depth with me. I had learned SR at university, read up on it, even going back to the original documents (Einstein and Feynman, Feynman because the light clock is sort of his). I had my uneasy feeling despite all this, and being told to go learn SR (again!) didn't really help.

Anyways, I've cast away a lot of my original stuff because I can now see that I was looking at the same thing as standard SR from a different perspective (I've not cast away everything, but I may in time cast away even the little that remains) and my deep-seated concerns that time dilation could actually be wrong were not justified.

However, if this feeling of there being something not quite right (which in my case were, as I said, deep-seated and may be equally concerning to others) is due to something as harmless as a pedagogical/psychological issue where some people intuitively think the way you do and others intuitively think another, yet both ways of thinking are completely valid, being just slightly different perspectives on the same thing, then it seems that there is some scope for improvement on how SR is taught.

I really do think that your suggestion a long time ago, when we had the discussion in which the diagram I posted here was central, was a good one.

You said that you would show your new students a similar diagram and explain that time dilation is not a TAFLC, and is not meant to be. I think you could go a little further and explain the physical significance of the actual TAFLC, and how it relates to length contraction so that c is invariant. That way, you would catch the people like me who feel that TAFLC is a useful equation and gently guide them towards a proper understanding of time dilation. At the same time, you would catch people like you, who go many years without grasping that there is any significance to a TAFLC.

Does that sound unreasonable?

cheers,

neopolitan

(I'm trying to get a lot into as few words as possible, it is late and it has been a long day. Sorry if there is anything which is hard to follow.)

 

[JesseM]

Thanks for that. With some things going on the background it took me some time to digest.

There is something which I find curious. It is a criticism of the pedagogy not of you nor of what time dilation is actually representing.

Note that you are "used to conceptualizing the standard and reversed time dilation equations as relating a clock time-interval on a clock at rest in the unprimed frame with a coordinate time-interval in the primed frame". It's quite a complex thing to internalise. When being taught, or trying to teach oneself, it is going to be a real uphill struggle to grasp that particular nature of the standard time dilation equation.

But as I said to Rasalhague, part of why this may seem "natural" and not really that complex if you've already spent some time studying other theories of physics like Newtonian mechanics or QM is that when solving physics problems, the usual convention is to pick some initial conditions, representing a frozen instant in which you can visualize the arrangement of all the parts of the system in space at that instant, and then evolve them forward in (coordinate) time using the dynamical equations. Once you've picked the frame, the coordinate time of that frame is something that you just get used to thinking of as "time" for the sake of solving that problem, almost like the absolute time of Newton, although in the background of your mind you know that it's frame dependent. Put it this way--if everyone still believed in absolute time and space (and therefore absolute velocity), and we knew there was a time dilation effect where clocks in motion relative to absolute space "ran slow" in an absolute sense, then would you still find it complex or difficult to internalize the notion of a time dilation equation that tells you how much real time goes by when a certain amount of ticks go by on a moving clock? And wouldn't this be pretty much analogous to wanting to know how much real space is taken up by a moving ruler whose marks indicate it's a certain length, but which is shrunk in absolute terms because it's moving? And in this case if you treat the frame corresponding to absolute space and time as the primed frame, and the frame of the moving clock and moving ruler as the unprimed frame, you get the usual equations $$\Delta t' = \Delta t * \gamma$$ and $$\Delta x' = \Delta x / \gamma$$. Of course you could also ask how many ticks go by on the moving clock given a certain amount of real time has passed, or what the rest length of a ruler is given that it's a certain length in real space, in this case you'd have to divide both sides of the first by gamma to get what I called the "reversed time dilation equation" $$\Delta t = \Delta t' /\gamma$$, and multiply both sides of the second by gamma to get $$\Delta x = \Delta x' * \gamma$$. On the other hand, if you're thinking in terms of absolute space and time and treating the absolute frame as the primed one, then the meaning of the TAFLC equation $$\Delta t' = \Delta t /\gamma$$ seems less intuitive to me; I guess it would come out to something like "given that two events on the worldline of a clock at rest in absolute space are separated by a coordinate time of $$\Delta t$$ in a frame moving at velocity v relative to absolute space, how much real time (or clock time, given that the clock is not slowed-down) passes between those two events?"

Anyway, if you can see my point that the time dilation and length contraction equation seem fairly "natural" in a universe with absolute space and time, then maybe you can see why, once a physics student has gotten used to the idea of picking a coordinate system and then taking that system's space and time coordinates for granted for the purposes of actual calculating the dynamical behavior of physical systems, then it might seem equally natural to ask how much coordinate time goes by when a certain amount of ticks go by on a moving clock (moving relative to that coordinate system), or how much coordinate space is taken up by a moving ruler. That's the best way I can think of to explain why the equations make intuitive sense to me, but obviously it's subjective so not everyone would have the same intuitions.

I do think there is a source of confusion there. I'd be willing to accept that it just me, but it seems there are many people with some problem or another with SR, which seems really odd. Why SR? Some are kooks for sure, but there are many people who seem to be otherwise able to maintain perfectly normal lives apart from an intuitive feeling that something is just not quite right about SR.

I've always thought that the main reason so many people have a problem with SR is because of the relativity of simultaneity, and what that might be taken to imply about the lack of any "objective" or "true" present, and therefore the lack of a real flow of time. In my experience--and I have seen a few physicists say the same thing--whenever people claim they have found a paradox in SR, the majority of the time it seems to come down to a failure to consider the relativity of simultaneity.

I won't say his name, and sadly he has probably passed away with cancer by now, but a professor in a city not far from where I lived up until recently was the lead lecturer for relativity at his university. He generously gave me many hours of his time to discuss my concerns and proposed solutions, and admitted that really, he didn't fully understand it. He did not stop me or explain that my concerns about time dilation were invalid, because he had never intuitively grasped the principles the standard way either.

Is it possible that, like DaleSpam said above, he just found it simpler to use the full Lorentz transform to approach any problem which compares different frames? Since all the other distinct equations like the time dilation equation, the length contraction equation, the relativity of simultaneity equation, and the velocity addition equation are all derived from the Lorentz transform, I can see the appeal of just using that one set of equations rather than bothering to keep track of a bunch of different ones dealing with different quantities and concepts in SR.

There is a fellow in southern Europe, another physics professor, albeit in a different field who expressed stronger views than I during our discussions that there was something amiss with SR. (I don't think SR is wrong but I do think it could be taught better.) A quantum physics professor in southern England also felt I was onto something with my arguments.

I have actually heard a few people working in quantum gravity who speculate that perhaps such a theory will restore a "true" flow of time and an objective present, Lee Smolin comes to mind for example. But I don't think this is a very common view.

I really do think that your suggestion a long time ago, when we had the discussion in which the diagram I posted here was central, was a good one.

You said that you would show your new students a similar diagram and explain that time dilation is not a TAFLC, and is not meant to be.

Yes, if I was ever actually in a position to be teaching a class on SR, I'd be sure to do that! I don't want people reading this to get the impression that I'm a professor or anything... ;)

I think you could go a little further and explain the physical significance of the actual TAFLC, and how it relates to length contraction so that c is invariant.

But when you say "how it relates to length contraction so that c is invariant", are you referring to the L/$$\Delta t$$ = c = L'/$$\Delta t'$$ argument? As I said that doesn't really make sense to me, because even if we assume that $$\Delta t$$ and $$\Delta t'$$ have the meaning given to them in the TAFLC equation, and L and L' have the usual meaning from the length contraction equation, I still don't see how it would make sense physically that L/$$\Delta t$$ and L/$$\Delta t'$$ could represent the "speed" of a single photon in two different frames if speed is given its usual interpretation as the distance covered in a certain interval of time. We can try going back to discussing this point if you want, or not if you don't want to get into it. As for the physical significance of the TAFLC, if we write it as $$\Delta t' = \Delta t / \gamma$$ I guess I would basically write it out as ""given that two events on the worldline of a clock at rest in the primed frame are separated by a coordinate time of $$\Delta t$$ in the unprimed frame, how much clock time (or coordinate time in the primed frame where the clock is at rest) passes between those two events?" Can you think of a more intuitive way to express the physical significance?

By the way, I'm about to go on a trip for a few days, so I probably won't be able to continue the discussion until next week sometime.

 

[neopolitan]

But when you say "how it relates to length contraction so that c is invariant", are you referring to the L/$$\Delta t$$ = c = L'/$$\Delta t'$$ argument? As I said that doesn't really make sense to me, because even if we assume that $$\Delta t$$ and $$\Delta t'$$ have the meaning given to them in the TAFLC equation, and L and L' have the usual meaning from the length contraction equation, I still don't see how it would make sense physically that L/$$\Delta t$$ and L/$$\Delta t'$$ could represent the "speed" of a single photon in two different frames if speed is given its usual interpretation as the distance covered in a certain interval of time. We can try going back to discussing this point if you want, or not if you don't want to get into it. As for the physical significance of the TAFLC, if we write it as $$\Delta t' = \Delta t / \gamma$$ I guess I would basically write it out as ""given that two events on the worldline of a clock at rest in the primed frame are separated by a coordinate time of $$\Delta t$$ in the unprimed frame, how much clock time (or coordinate time in the primed frame where the clock is at rest) passes between those two events?" Can you think of a more intuitive way to express the physical significance?

That definition is correct, although I would imagine a new student would need to be eased into it.

I can see why you can't make sense of L/t = c = L'/t'.

You specifically want to measure a time interval between two events in the primed frame and then compare that to a time inverval in the unprimed frame.

I wasn't doing that. I was saying that any time inverval in the primed frame between two events which are colocal in the primed frame, will be shorter in the unprimed frame than two analogous (but not the same) events in the unprimed frame. The half-life of one muon in the primed frame (viewed from the primed frame) will be the same as the half-life of a totally different muon in the unprimed frame (viewed from the unprimed frame. (Yes, I know half-lives are statistical, but using a gross misrepresentation here might still be instructive.)

What I am saying is that the half-life of the muon in the primed frame (viewed from the primed frame) will be less than the half-life of the muon in the primed frame (viewed from the unprimed frame).

In the example BobS raised earlier in the thread, a muon at a gamma of 29.3 had a measured life time of 64.4ms as opposed to a normal (gamma of 1) life time of 2.2ms.

In the experiment he refers to, I would call the measured lifetime t and I could use the gamma to calculate what the life time in muon's "rest frame" was (quotation marks because "rest frame" is a bit of a misnomer under the circumstances). I'd prime the rest frame of the muon and leave the laboratory rest frame unprimed. That would give me:

t' = t/gamma = 64.4ms / 29.3 = 2.2ms

If I had a different experiement, using light clocks, this is how I would be doing it.

At rest in the laboratory, my light clock has a tick time of 2.2ms. That makes the distance between mirrors ct/2 = 330km (giving a L = 660km, the total distance a photon travels between ticks).

Conceptually, put the light clock at a gamma factor of 29.3 (in reality, this would prove difficult).

I will measure, in the laboratory, that the time between ticks of the light clock is now 64.4ms.

This 64.6ms is the t which is equivalent to the t from the muon example. It is not equivalent to the t which I used in ct/2 = 330km (that t was 2.2ms).

What I do know is that, in the laboratory's frame, the photon in the light clock has not traveled 330km in 64.4ms. As you showed before (using time dilation) the photon has to travel much further from one mirror to the other mirror in one direction and a bit less in the other direction.

So the distance traveled between ticks (in the laboratory) is not the same L as before but rather ct where t = 64.4ms ... eg, 19320km.

This L, divided by this t = 19320km/64.4ms = 300000 km/s

The distance traveled in the rest frame of the light clock is the old L (330km) and the time a photon takes to travel between them and back again is the old t (2.2ms).

This L, divived by this t = 660km/2.2ms = 300000 km/s

If you want to use the clock in the laboratory you as your reference point, you have to do this:

While a photon in the laboratory moves between mirrors, traveling 660km in 2.2ms - what happens to a photon which is at gamma of 29.3?

If 2.2ms has elapsed in the laboratory, then a period of 2.2ms/29.3 = 75μs will have elapsed in the rest frame of the test clock (the one accelerated to gamma of 29.3). The test clock will not have ticked. In the rest frame of the test clock, the test clock's photon will only have traveled a distance of 22.5 km.

This time and this distance are t' and L'.

L'/t' = 22.5km/75μs = 300000 km/s

I hope this helps.

cheers,

neopolitan

 

[Rasalhague]

Anyway, if you can see my point that the time dilation and length contraction equation seem fairly "natural" in a universe with absolute space and time, then maybe you can see why, once a physics student has gotten used to the idea of picking a coordinate system and then taking that system's space and time coordinates for granted for the purposes of actual calculating the dynamical behavior of physical systems, then it might seem equally natural to ask how much coordinate time goes by when a certain amount of ticks go by on a moving clock (moving relative to that coordinate system), or how much coordinate space is taken up by a moving ruler. That's the best way I can think of to explain why the equations make intuitive sense to me, but obviously it's subjective so not everyone would have the same intuitions.

Maybe this is part of what confuses a novice like me with less experience of physics in general to draw on. Namely that, having been taught that there is no absolute space and time, we're then tacitly invited to pretend there is “for the sake of solving the problem”. But as a beginner, that leaves you wondering *how much* of the your intuition you're allowed to hang onto in this particular exercise. And unless it’s made explicit, you just can't tell, because the one thing you've been warned is that you can't trust your intuition when it comes to relativity. So I worry that I might make mistakes by being lulled by such a natural-seeming way of conceptualising it. Or, to put it another way, the "natural" way of treating one frame as preferential, for the sake of convenience, can sometimes feel to me as if it's bringing swarming after it all those apparent paradoxes that disappear only when you abandon certain intuitions, such as--especially--absolute simultaneity. But maybe when I'm more familiar with SR, that won't be so much of an issue.

I suppose "time dilation" and "length contraction" being just a shorthand for the full Lorentz transformation, of use in a special cases, the thing to be learned is what those special cases are, and (on a more philosophical or abstract level) why a different special case is thus highlighted for time from the special case thus highlighted for space. Regarding which, I've found this a fascinating discussion.

I've always thought that the main reason so many people have a problem with SR is because of the relativity of simultaneity, and what that might be taken to imply about the lack of any "objective" or "true" present, and therefore the lack of a real flow of time. In my experience--and I have seen a few physicists say the same thing--whenever people claim they have found a paradox in SR, the majority of the time it seems to come down to a failure to consider the relativity of simultaneity.

Definitely! I certainly found that when I learned that simultaneity was relative too--although it's such a fiendishly counterintuitive concept--that was the moment when some of these bizarre ideas first started to fall into place. They're still very hard for me to understood, but they no longer feels an affront to reason! The other technique that often clears things up for me is to break the problem down and think of it in terms of events. That often helps to root out the false assumptions lurking in my brain.

As for the physical significance of the TAFLC, if we write it as $$\Delta t' = \Delta t / \gamma$$ I guess I would basically write it out as "given that two events on the worldline of a clock at rest in the primed frame are separated by a coordinate time of $$\Delta t$$ in the unprimed frame, how much clock time (or coordinate time in the primed frame where the clock is at rest) passes between those two events?" Can you think of a more intuitive way to express the physical significance?

As in my example, I found it helpful to give the notional observers names, and make their circumstances perfectly symmetrical. That seemed to be the only way I could start get my head around which parts of the problem were significant features of spacetime geometry, and which were accidental details of the example. In the descriptions I'd encountered, I often found myself struggling to keep track over whether a particular author was using the primed/unprimed convention to represent some unique feature of a particular frame. Some introductions use unprimed as you describe, but others use it according to some other convention, or arbitrarily. And of course, where the problem is more elaborate and involves converting back and forth between frames, or where the direction of movement is significant, it's less obvious which frame is the more natural choice to be called unprimed.

 

[JesseM]

That definition is correct, although I would imagine a new student would need to be eased into it.

I can see why you can't make sense of L/t = c = L'/t'.

You specifically want to measure a time interval between two events in the primed frame and then compare that to a time inverval in the unprimed frame.

I wasn't doing that. I was saying that any time inverval in the primed frame between two events which are colocal in the primed frame, will be shorter in the unprimed frame than two analogous (but not the same) events in the unprimed frame. The half-life of one muon in the primed frame (viewed from the primed frame) will be the same as the half-life of a totally different muon in the unprimed frame (viewed from the unprimed frame. (Yes, I know half-lives are statistical, but using a gross misrepresentation here might still be instructive.)

What I am saying is that the half-life of the muon in the primed frame (viewed from the primed frame) will be less than the half-life of the muon in the primed frame (viewed from the unprimed frame).

OK, agree with you so far.

In the example BobS raised earlier in the thread, a muon at a gamma of 29.3 had a measured life time of 64.4ms as opposed to a normal (gamma of 1) life time of 2.2ms.

In the experiment he refers to, I would call the measured lifetime t

OK, as long as you are aware that here you are using the reverse of the "normal" convention, which is to use the unprimed frame for the rest frame of the "clock" (in this case the natural clock provided by the muon's decay) and the primed frame the frame where we are measuring the time interval between events on the worldline of a moving clock. If you want to reverse this and call the muon's rest frame the primed frame, then the "normal" time dilation equation would be written as $$\Delta t = \Delta t' * \gamma$$, the "reversed time dilation equation" would be written as $$\Delta t' = \Delta t / \gamma$$, and the TAFLC would be written as $$\Delta t = \Delta t' / \gamma$$.

and I could use the gamma to calculate what the life time in muon's "rest frame" was (quotation marks because "rest frame" is a bit of a misnomer under the circumstances). I'd prime the rest frame of the muon and leave the laboratory rest frame unprimed. That would give me:

t' = t/gamma = 64.4ms / 29.3 = 2.2ms

Yes. But just to be clear about terminology, do you agree that this is not the TAFLC, but just the reversed version of the regular time dilation equation?

If I had a different experiement, using light clocks, this is how I would be doing it.

At rest in the laboratory, my light clock has a tick time of 2.2ms. That makes the distance between mirrors ct/2 = 330km (giving a L = 660km, the total distance a photon travels between ticks).

Conceptually, put the light clock at a gamma factor of 29.3 (in reality, this would prove difficult).

I will measure, in the laboratory, that the time between ticks of the light clock is now 64.4ms.

This 64.6ms is the t which is equivalent to the t from the muon example. It is not equivalent to the t which I used in ct/2 = 330km (that t was 2.2ms).

What I do know is that, in the laboratory's frame, the photon in the light clock has not traveled 330km in 64.4ms. As you showed before (using time dilation) the photon has to travel much further from one mirror to the other mirror in one direction and a bit less in the other direction.

So the distance traveled between ticks (in the laboratory) is not the same L as before but rather ct where t = 64.4ms ... eg, 19320km.

This L, divided by this t = 19320km/64.4ms = 300000 km/s

Yes.

The distance traveled in the rest frame of the light clock is the old L (330km) and the time a photon takes to travel between them and back again is the old t (2.2ms).

This L, divived by this t = 660km/2.2ms = 300000 km/s

For clarity we can call this distance in the light clock rest frame L' = 660 km and this time t' = 2.2 ms so it maps to your L/t = c = L'/t', correct? In this case, do you agree that t and t' are related not by the TAFLC but by the standard time dilation equation (written with your unusual convention of labeling the clock rest frame as the primed frame) t = t' * gamma? And do you also agree that L and L' are related not by the length contraction equation but by an equation which looks like the "spatial analogue of time dilation" (although I'm not sure L and L' can be assigned the same physical meaning) L = L' * gamma?

As long as you agree with this stuff I have no problem with the L/t = c = L'/t' argument, but I thought you had been saying that the TAFLC was the equation that was useful in understanding the invariance of c, not the time dilation equation. I guess if you want to say that the equation L = L' * gamma is useful for understanding the invariance of the speed of light that would have some truth, although I think this only works when you're talking about the two-way speed away from some fixed point in the clock's frame and back, and as I said I don't know if the physical meaning of L and L' here can be mapped to the "spatial analogue of time dilation" equation even though it looks the same.

Finally, you said earlier: "You specifically want to measure a time interval between two events in the primed frame and then compare that to a time inverval in the unprimed frame. I wasn't doing that." It seems to me you are doing that, with the two events being 1) the event of the photon leaving the bottom mirror of the light clock which is moving in the lab frame, and 2) the event of the photon returning to the bottom mirror of that same clock. The time between these events is t' = 2.2 ms in the clock rest frame and t = 64.4 ms in the lab frame. The part I had not understood was that you were not using L and L' to represent the distance between these events in the two frames, but rather the total distance covered by the photon in each frame between these two events; this would be identical to the distance between the events if the events were on a single straight photon worldline, but since you are talking about the two-way speed of light rather than the one-way speed of light, the photons are reflected so their worldlines aren't straight.

If you want to use the clock in the laboratory you as your reference point, you have to do this:

While a photon in the laboratory moves between mirrors, traveling 660km in 2.2ms - what happens to a photon which is at gamma of 29.3?

If 2.2ms has elapsed in the laboratory, then a period of 2.2ms/29.3 = 75μs will have elapsed in the rest frame of the test clock (the one accelerated to gamma of 29.3).

No, 75 microseconds would represent how much time has elapsed on the test clock (if the test clock had closer mirrors so it could show time-intervals that small) in 2.2 ms of time in the lab frame. In the test clock's own frame, it's the lab clock that's running slow relative to the test clock, so when the lab clock has ticked forward 2.2 ms, the test clock has ticked forward 64.4 ms.

 

[neopolitan]

There still seems to be some confusion.

We talk about L as it has to be a ruler or a rod or a length. They are convenient devices, but L could be a distance between two randomly selected points in a rest frame (I want to say my rest frame, but it can be any rest frame).

We talk about t as if it has to be attached to events, like ticking of a clock, or formally defined events. But t could be the time interval between two randomly selected times.

We can imagine putting two pins on a map and measuring the distance. We have difficulty putting two pins in time and measuring the temporal distance. But I take TAFLC as being for measuring between these two pins in time, in the same was a LC is for measuring between two pins on the map. We take a different perspective on them by putting us and pins into different inertial frames.

If I get myself an inertial frame where two time pins are in the same position, then they will be as far apart in time as they can be. If I get myself an inertial frame where the two length pins are simultaneous, then they will be as far apart in length as they can be.

But, assuming all the pins are in the same frame (ie they share a frame in which the time pins have zero length separation and the length pins have zero time separation), then from any other frame: t' = t/gamma and L'=L/gamma where t and L are the maximum time and length separations for the respective pins.

I'm deliberately using a different approach.

cheers,

neopolitan

 

[neopolitan]

No, 75 microseconds would represent how much time has elapsed on the test clock (if the test clock had closer mirrors so it could show time-intervals that small) in 2.2 ms of time in the lab frame. In the test clock's own frame, it's the lab clock that's running slow relative to the test clock, so when the lab clock has ticked forward 2.2 ms, the test clock has ticked forward 64.4 ms.

Simultaneity issues here. I was only taking the lab's perspective, one perspective at a time. But yes, you can reverse it around, and due to relativity of simultaneity, the muon will decay in its own frame while the lab clock reads 75 microsecond (but since we can't be muons anymore than we can be photons, it makes sense to use the lab perspective. I don't think that particle physicists would report that the muon decays after 75 microseconds on the lab clock when read from the muon's own frame in which a clock, if it could be accelerated to a gamma of 29.3, would read 2.2 ms).

 

[Mentz114]

(but since we can't be muons anymore than we can be photons, it makes sense to use the lab perspective.

Muons have mass, so we can 'be muons'. I thought you might like to know this.

 

[neopolitan]

Muons have mass, so we can 'be muons'. I thought you might like to know this.

Hm, if you were a muon, you wouldn't be one for long.

However, I don't think there is much difference between "I can't be a northern polka dotted, orange bellied, bearded unicorn" and "I can't be a ballet lady". I think there are a few good reasons why I can't be a muon (even though a muon has mass). Equally, I don't think that not having mass is the only thing preventing me from being a photon.

Perhaps I missed a key lecture at uni.

(There's another feeble attempt at sarcasm )

PS Have you got a thing about muons? It's just that you have only popped your head into make comment about them. If they are off limits or something, just let me know and I will use another example.

 

[JesseM]

Damn, I edited this post when I meant to reply to it to add a small comment, hence erasing everything else but the small new comment...I'll have to try to reconstruct it.

 

[Rasalhague]

If I get myself an inertial frame where two time pins are in the same position, then they will be as far apart in time as they can be. If I get myself an inertial frame where the two length pins are simultaneous, then they will be as far apart in length as they can be.

But, assuming all the pins are in the same frame (ie they share a frame in which the time pins have zero length separation and the length pins have zero time separation), then from any other frame: t' = t/gamma and L'=L/gamma where t and L are the maximum time and length separations for the respective pins.

That's a very neat summary. It brings out very clearly where the symmetry lies (between time and space), and where the difference lies (between (1) frames in which a timelike separation has no space component, or frames in which a spacelike separation has no time component, and (2) other frames in which the separation, timelike or spacelike, has a mixture of time and space coordinates). Finger's crossed I've got the terminology corrent there...

 

[Rasalhague]

That's a very neat summary. It brings out very clearly where the symmetry lies (between time and space), and where the difference lies (between (1) frames in which a timelike separation has no space component, or frames in which a spacelike separation has no time component, and (2) other frames in which the separation, timelike or spacelike, has a mixture of time and space coordinates). Finger's crossed I've got the terminology corrent there...

Ah, I just read Jesse's reply after I posted this. I see the point about it being the minimum separation. Taking that into account, it does still seem a satisfying way of looking at it.

 

[Rasalhague]

"given that two events on the worldline of a clock at rest in the primed frame are separated by a coordinate time of $$\Delta t$$ in the unprimed frame, how much clock time (or coordinate time in the primed frame where the clock is at rest) passes between those two events?" Can you think of a more intuitive way to express the physical significance?

Is this equivalent to saying: "Two events are separated by a timelike interval $$\Delta \tau$$. In frame [tex]S[tex], this separation has a time component $$\Delta t > \Delta \tau$$. Given the value of $$\Delta t$$, how can we calculate $$\Delta \tau$$? Answer: $$\Delta \tau = \Delta t / \gamma$$. The inverse question being: "Given the value of $$\Delta \tau$$, how can we calculate $$\Delta t$$? Answer: $$\Delta t' = \Delta \tau * \gamma$$.

Alternatively:

Given two events $$E_{a1}$$ and $$E_{a2}$$, colocal in some frame $$S$$, with (time) interval $$\Delta t$$, what is the (time) interval $$\Delta t'$$ in some other frame, moving at constant velocity $$u$$ relative to $$S$$, between two events $$E_{b1}$$ and $$E_{b2}$$, colocal in $$S'$$, if $$t_{a1}$$ = $$t_{b1}$$, and $$t_{a2}$$ = $$t_{b2}$$?

$$\Delta t' = \Delta t / \gamma$$.

As opposed to time dilation:

Given two events $$E_{a1}$$ and $$E_{a2}$$, colocal in some frame $$S$$, with (time) interval $$\Delta t$$, what is the (time) interval $$\Delta t'$$ in some other frame $$S'$$, moving at constant velocity $$u$$ relative to $$S$$, between two events $$E_{b1}$$ and $$E_{b2}$$, colocal in S', if $$t'_{a1}$$ = $$t'_{b1}$$, and $$t'_{a2}$$ = $$t'_{b2}$$?

$$\Delta t' = \Delta t * \gamma$$.

So would it be fair to say that there really is no fundamental or physical difference between "reverse time dilation" and "temporal analogue of length contraction" ("time contraction")? They ask the same question, only with different names given to the frames. If the problem you're working on only involves one question, or if it only involves asking one type of question of one frame, and the other type of question of the other frame, then you can avoid ever having to use the form $$\Delta t' = \Delta t / \gamma$$, and instead always use $$\Delta t = \Delta t' / \gamma$$. But if you want to ask both types of question in both directions, then you'd have to use $$\Delta t' = \Delta t / \gamma$$, wouldn't you? Or else swap over the labels you've given to the frames as the occasion demands.

 

[Rasalhague]

The garbled text in my previous post should have read:

Is this equivalent to saying: "Two events are separated by a timelike interval $$\Delta \tau$$. In frame $$S$$, this separation has a time component $$\Delta t > \Delta \tau$$. Given the value of $$\Delta t$$, how can we calculate $$\Delta \tau$$? Answer: $$\Delta \tau = \Delta t / \gamma$$. The inverse question being: "Given the value of $$\Delta \tau$$, how can we calculate $$\Delta t$$? Answer: $$\Delta t' = \Delta \tau * \gamma$$.

 

[Rasalhague]

Actually, writing it out in these terms and then thinking about how I'd write out the TAFLC equation in words makes me realize that the question of whether there's really any difference between the TAFLC equation and the reversed time dilation equation is actually rather subtle. If you look at my diagram, you see that the TAFLC isn't really giving you the time-interval between any pair of pink events at all, since none of the events are at the position of the top of the double-headed arrow that I use to represent the delta-t' of the TAFLC equation; it's only if you were to draw a new pair of events that are colocated in the primed frame, at the top and bottom of that double-headed arrow on that diagram, that the TAFLC would tell you the same think about the time between those new events in both frames that the reversed time dilation equation tells you about the time between the colocated events in the primed frame. So I guess what that would mean conceptually is that if you choose your pair of events at the start, then the time dilation equation + reversed time dilation equation tell you everything you need about the relation between the time intervals connecting those specific events in your two frames (one of which must be the frame where they're colocated). In this context the TAFLC equation is actually not telling you about the time-interval between those specific events in either frame, although you could of course draw in some new events such that the times delta-t and delta-t' in the TAFLC equation had the same meaning for that new pair of events that the times delta-t and delta-t' in the time dilation (and reversed time dilation) equation have for the original pair of events. But then if you want to talk about the time between the new pair, why not just start over and have them be the starting events? I guess conceptually what I would say is that to use any of these time equations you should always be clear on what two events you're interested in at the start, and once you've picked them then it's the time dilation and reversed time dilation equation that tell you the relation between the time-intervals in both frames, while the TAFLC is telling you something more abstract about the time in the non-colocated frame between planes of simultaneity from the colocated frame that pass through both events.

But couldn't you look at the conventional time dilation equation in a similar way? In each case you want to know something about the timing of two events. You specify something about the events which you want information about (which other events they have to be simultaneous with, and according to whose definition of simultaneity), in both cases without knowing exactly which events you're looking for, and the equations tell you. It could well be that I'm missing the subtlety though. I need to read these posts more carefully and think this over.

 

[JesseM]

OK, here's the recreation of the last post I accidentally edited away:

There still seems to be some confusion.

We talk about L as it has to be a ruler or a rod or a length. They are convenient devices, but L could be a distance between two randomly selected points in a rest frame (I want to say my rest frame, but it can be any rest frame).

We talk about t as if it has to be attached to events, like ticking of a clock, or formally defined events. But t could be the time interval between two randomly selected times.

We can imagine putting two pins on a map and measuring the distance. We have difficulty putting two pins in time and measuring the temporal distance. But I take TAFLC as being for measuring between these two pins in time, in the same was a LC is for measuring between two pins on the map. We take a different perspective on them by putting us and pins into different inertial frames.

But look at my diagram again. If the pink dots are the pins, with two colocated in the unprimed frame and two simultaneous in the unprimed frame, then it is actually the time dilation equation that compares the time in the two frames between the events that are colocated in the unprimed frame, and the "spatial analogue for time dilation" (SAFTD) equation that compares the distances in the two frames between the events that are simultaneous in the unprimed frame. The TAFLC equation doesn't tell you the time between any pair of pink events in the diagram, although you could invent a new pair of events such that it would--these new events would have to be colocated in the primed frame.

If I get myself an inertial frame where two time pins are in the same position, then they will be as far apart in time as they can be.

Actually that's backwards, the time between events is minimized in the frame where they're at the same position. Suppose I have been moving inertially my whole life, and one event is the event of my birth while the other is the event of my turning 30. The time between these events is 30 years in the frame where I am at rest and they occur at the same location, but in a frame where I am moving there is a greater time between the events because I am aging more slowly.

If I get myself an inertial frame where the two length pins are simultaneous, then they will be as far apart in length as they can be.

That's not quite correct either. If you want to analyze length contraction in terms of just two events rather than three (in the case of three, #1 would be an event on the worldline of the object's left end, #2 would be an event on the worldline of the object's right end that's simultaneous with #1 in the object's rest frame, and #3 would be an event on the worldline of the right end that's simultaneous with #1 in the frame where the object is moving), then you have to pick two events on the worldline of either end of the object that are simultaneous in the frame where the object is moving, but non-simultaneous in the object's rest frame (since both ends of the object have a constant position in the object's rest frame, events on either end will still be separated by the rest length L even if they aren't simultaneous). The distance between these events will be greater in the object's rest frame where they're non-simultaneous (because rest length is greater than moving length), so they aren't at a maximal separation in the frame where the events are simultaneous. In fact it turns out that events will actually have a minimal spatial distance in the frame where they are simultaneous, you can see this by considering the more general equation for the separation between events in two arbitrary frames:

$$\Delta x' = \gamma (\Delta x - v \Delta t)$$

If you choose the unprimed frame to be the one where they're simultaneous, then $$\Delta t$$ = 0 so you're left with $$\Delta x' = \gamma * \Delta x$$, which shows that the distance is always greater in the non-simultaneous frame.

Aside from these caveats, I agree with the idea that you can define the meaning of the two frames in equations like time dilation by first picking two events and then making clear which is supposed to be the frame where they are colocated (if they are timelike-separated) or which is supposed to be the frame where they are simultaneous (if spacelike-separated). Writing it out in words, the standard time dilation equation would be:

(time between events in frame where they are not colocated) = (time between events in frame where they are colocal) * gamma

Likewise, the reversed time dilation equation would be:

(time between events in frame where they are colocal) = (time between events in frame where they are not colocated) / gamma

Thinking about writing it in words, it may seem a bit subtle to say what the difference is between the TAFLC equation and the reversed time dilation equation. As I said, if you look at my diagram you see that the double-headed arrow representing the dt' in the TAFLC does not have any of the three pink events at the top end of it; you would have to invent a new pair of events at either end of this double-headed arrow in order to phrase the TAFLC in terms of time intervals between events, and in that case you would write it exactly like the reversed time dilation equation above, except with the understanding that you were now referring to that new pair of events. So the way I would conceptualize this situation is to say that in order to talk about any of these equations, you first have to specify a single pair of events you want to talk about, and then in terms of those specific events the time dilation and reversed time dilation equations tell you everything you want to know about the time interval between the events in two frames (one of which is the one where they're colocated), whereas in terms of those events the TAFLC is telling you something more abstract about the time-interval (in the frame where the events are not colocated) between surfaces of simultaneity from the the frame where the events are colocated. Of course you could start with a new pair of events so that the time interval given by the TAFLC applied to the previous events is just the time interval between the new events in the frame where they're colocated, but then you're really talking about the reversed time dilation equation for these new pair of events, not the TAFLC for them.

But, assuming all the pins are in the same frame (ie they share a frame in which the time pins have zero length separation and the length pins have zero time separation), then from any other frame: t' = t/gamma and L'=L/gamma where t and L are the maximum time and length separations for the respective pins.

As I said above, t and L should be the minimum time and distance separation for the pins, there is no upper limit on their separations (there is an upper limit on the length of a physical object when viewed in different frames, but the concept of the length of an object in different frames is quite different from the concept of the spatial distance between a pair of events in different frames). And if the unprimed frame is the one where the time pins are colocated and the space pins are simultaneous, then the equations above are incorrect, they should be t' = t*gamma and L' = L*gamma, representing the standard time dilation equation along with the SAFTD equation. Do you disagree?

 

[JesseM]

Ah, I just read Jesse's reply after I posted this. I see the point about it being the minimum separation. Taking that into account, it does still seem a satisfying way of looking at it.

Also see the points I made in the re-created version of that post (the original of which I accidentally deleted) about the differences between the concept of the length of a physical object in different frames vs. the concept of the distance between a pair of events in different frames. Even though the length of an object is maximized in its rest frame, the distance between a pair of events is minimized in the frame where they are simultaneous.

 

[Mentz114]

neopolitan;
no, I don't have a thing about muons. I did not introduce the subject so your comment makes no sense. You give the impression that muons can't have a frame of reference, in which you are wrong. I'm trying to shine some light here into your fog of misunderstanding, and you respond with insults and sarcasm.

I enjoyed your little biog about talking to people ( Professors even ) about your doubts and problems with relativity. I hope you get cured soon because it's costing some people an awful lot of effort.

M

 

[JesseM]

Is this equivalent to saying: "Two events are separated by a timelike interval $$\Delta \tau$$.

OK, that would be equivalent to the proper time along the worldline of an inertial object that goes from one event to the other, which of course is the same as the coordinate time between the events in that object's rest frame, where the events occur at the same coordinate position.

In frame $$S$$, this separation has a time component $$\Delta t > \Delta \tau$$. Given the value of $$\Delta t$$, how can we calculate $$\Delta \tau$$? Answer: $$\Delta \tau = \Delta t / \gamma$$. The inverse question being: "Given the value of $$\Delta \tau$$, how can we calculate $$\Delta t$$? Answer: $$\Delta t = \Delta \tau * \gamma$$.

Yes, although your "inverse question" corresponds to the normal time dilation equation (with the most common notation being to use a primed t' where you've used an unprimed t, and an unprimed t where you've used $$\tau$$), whereas your first question corresponds to what I've called the "reversed time dilation equation" (where you just divide both sides of the normal time dilation equation by gamma).

Alternatively:

Given two events $$E_{a1}$$ and $$E_{a2}$$, colocal in some frame $$S$$, with (time) interval $$\Delta t$$, what is the (time) interval $$\Delta t'$$ in some other frame, moving at constant velocity $$u$$ relative to $$S$$, between two events $$E_{b1}$$ and $$E_{b2}$$, colocal in $$S'$$, if $$t_{a1}$$ = $$t_{b1}$$, and $$t_{a2}$$ = $$t_{b2}$$?

$$\Delta t' = \Delta t / \gamma$$.

Since you wrote $$t_{a1}$$ = $$t_{b1}$$ rather than $$t'_{a1}$$ = $$t'_{b1}$$, I take it you want these events to be simultaneous in the unprimed frame rather than the primed frame? If so, then if we want to conceptualize this in terms of the coordinate time in two frames between a single pair of events as in neopolitan's formulation, then we're really talking about the second pair of events $$E_{b1}$$ and $$E_{b2}$$ here; we know the time between them in the unprimed frame, and want to know the time between them in the primed frame where they are colocated. So, this would indeed be the "reversed time dilation equation" you have above, but it would be the opposite of the usual convention about primed and unprimed (the usual convention being that the frame in which the two events are colocated would be the unprimed one).

As opposed to time dilation:

Given two events $$E_{a1}$$ and $$E_{a2}$$, colocal in some frame $$S$$, with (time) interval $$\Delta t$$, what is the (time) interval $$\Delta t'$$ in some other frame $$S'$$, moving at constant velocity $$u$$ relative to $$S$$, between two events $$E_{b1}$$ and $$E_{b2}$$, colocal in S', if $$t'_{a1}$$ = $$t'_{b1}$$, and $$t'_{a2}$$ = $$t'_{b2}$$?

$$\Delta t' = \Delta t * \gamma$$.

Yes, although if we think in terms of a single pair of events as before, here you've reversed the convention about which frame is the one where they're colocated.

So would it be fair to say that there really is no fundamental or physical difference between "reverse time dilation" and "temporal analogue of length contraction" ("time contraction")? They ask the same question, only with different names given to the frames.

I don't think so--as I said in my post to neopolitan, if you think in terms of starting with a pair of events and then asking various questions about time-intervals involving those specific events, then the TAFLC equation is really asking something more like "in the frame where the events are not colocated, what is the temporal separation between two surfaces of simultaneity from the frame where they are colocated, given that each surface passes through one of the two events?" But this point about starting with a single pair of events brings me to your next post where you were responding to a similar comment from the post I accidentally deleted:

But couldn't you look at the conventional time dilation equation in a similar way? In each case you want to know something about the timing of two events. You specify something about the events which you want information about (which other events they have to be simultaneous with, and according to whose definition of simultaneity), in both cases without knowing exactly which events you're looking for, and the equations tell you. It could well be that I'm missing the subtlety though. I need to read these posts more carefully and think this over.

I think you always have to know what the events are physically, like particular readings on a physical clock, or any other observed events you like, and are then interested in saying various things relating to how different coordinate systems view them, like the difference in coordinate time between the events or which readings on a different physical clock are simultaneous with these events in a particular frame (and what the difference is between the two readings on that clock). I suppose you can ask questions in such a way that you don't know both events in advance, like "which reading on this clock occurs at a time interval of $$\Delta t$$ after the clock reading 0 in my frame", but for the question to be well-defined it must uniquely determine the events in question even if you don't know them until you do some calculations.

If the problem you're working on only involves one question, or if it only involves asking one type of question of one frame, and the other type of question of the other frame, then you can avoid ever having to use the form $$\Delta t' = \Delta t / \gamma$$, and instead always use $$\Delta t = \Delta t' / \gamma$$. But if you want to ask both types of question in both directions, then you'd have to use $$\Delta t' = \Delta t / \gamma$$, wouldn't you? Or else swap over the labels you've given to the frames as the occasion demands.

But what do you mean by "both types of questions"? What events are you asking questions about? If you're asking about more than a single pair of events then in that case I'd agree you might use both of those equations to talk about time intervals between events, but since you're no longer talking about a single pair of events you'd have to have some different notation to distinguish between verbal formulations like "time-interval in the unprimed frame between events A and B" and "time-interval in the unprimed frame between events C and D"--perhaps you could use $$\Delta t_{AB}$$ and $$\Delta t_{AC}$$ in this case. Then if A and B are colocated in the primed frame while C and D are colocated in the unprimed frame, you might write $$\Delta t'_{AB} = \Delta t_{AB} / \gamma$$ along with $$\Delta t_{CD} = \Delta t'_{CD} / \gamma$$, but I would refer to the first as "the reversed time dilation equation for events A and B" and the second as "the reversed time dilation equation for events C and D", in words they would both come out to:

(time between specified events in frame where they are colocated) = (time between specified events in frame where they are not colocated) / gamma