Is it time to "retire" time dilation and length contraction?

[SlowThinker]

It seems, until someone invents a hyperbolic, rather than Euclidean, paper, people will be confused...

Maybe it's possible to use this trick:
First demonstrate Euclidean distance using a circle.
Draw a line A-B at an angle, and ask "How do I measure its length?" Then draw a circle centered at A going through B, and on that circle mark points H to the right of A and V above A. Then both A-H and A-V are the length.

Then switch to Minkowski distance using a hyperbola.
Draw the asymptotes through A, and then a hyperbola through B, marking either X or T, depending on the angle of A-B. Then you have the way to show Minkowski distance. And you get "timelike", "spacelike" and "lightlike" for free.

Now you can draw the Twin paradox on a blackboard and it's clear that the traveling twin travels shorter Minkowski distance.

 

[PeterDonis]

obviously the length of the lines in the diagram does not have any physical meaning, as long as the lines do not reunite.

The Euclidean lengths don't have meaning even if the lines do reunite. The third line in the triangle given in my last post ends up being spacelike, and we want to restrict attention to timelike lines, so instead consider the triangle described by the three points A, B, C given by: (x, t) = $(0, 0)$, $(0, 0.25)$, $(\sqrt{3}, 2)$. The side lengths are AB = 0.25, AC = 1, BC = 0.25. So the spacetime length of BC is exactly the same as that of AB, yet the Euclidean length of BC is much longer than that of AB--in fact it's nearly as long as that of AC. So we have a total spacetime length of 0.5, AB + BC, represented by Euclidean lines that, taken together, are almost the same length as a Euclidean line, AC, that represents twice the total spacetime length.

 

[Smattering]

No, this is wrong. The lines do have a physical meaning as the proper time elapsed from the start of the line to the end of the line. This is independent of reuniting with a different line (or even the existence of a different line).

We were talking about the length of the diagram lines within the Euclidean medium here. Not about the world lines.

 

[Orodruin]

We were talking about the length of the diagram lines within the Euclidean medium here. Not about the world lines.

In that case it is still wrong. The Euclidean length never has physical meaning even when the curves reunite.

 

[PWiz]

I just looked back at post #35, and I see that there are (major) problems with it.

1) I've used "length" to describe the length of a worldline in a spacetime diagram in some parts of the post, while in some other parts I've used it to describe the "distance on the manifold" (which is represented by a family of hyperbolas in a spacetime diagram) without explicitly mentioning what I mean. Unfortunately, this is giving the impression that I believe that the Euclidean "length" of a worldline as calculated in a spacetime diagram has physical significance. This is not the case - I'm well aware that the spacetime diagram "length" is useless. Taking the 1+1 dimensional case for the spacetime diagram, the "length" of a worldline on the diagram will be given by $s = \int \sqrt{1+ \left ( \frac{dx}{dt} \right ) ^2 }dt$. This has no relation with the spacetime interval (in flat 1+1 spacetime): $s = \int \sqrt{1- \left ( \frac{dx}{dt} \right ) ^2 }dt$ (I've used the ---+ convention because its much more convenient to use to calculate the line element on the manifold for timelike intervals, as $ds^2 < 0$ in the +++- convention for such intervals, and we have to unnecessarily add a -ve sign in the integral to calculate $|s|$ in that case.) When I said

But the "length" of this route is defined in a way different from your intuition: ds2=dxidxj−dt2ds^2 = dx^i dx^j - dt^2 (dxidxjdx^i dx^j is the ordinary spatial distance between the two events).

I meant the line element on the manifold here, and I didn't mention what I was talking about in the original post - the Euclidean spacetime diagram length or the line element on the manifold. (A fatal flaw)

In fact, the spacetime diagram length and the actual spacetime interval can never be related even in non-flat spacetime, because the Euclidean metric tensor in the spacetime diagram (used for measuring the Euclidean length of a curve in the diagram) will always be positive-definite while the (pseudo) metric tensor of spacetime will always have a Lorentzian signature.
If we solve the Euler-Lagrange equation for the spacetime diagram "length", we get the function $x=At + B$ which acts as the extremizer, where $A$ and $B$ are constants. This function is a minimizer here, so $v =$ constant is going to result in the "shortest" worldline on a spacetime diagram. This is what I meant when I said

It is the way you think - the accelerated twin has a "longer" worldline than the unaccelerated twin. Generalizing Newton's first law, we can say that the unaccelerated twin moves on a geodesic by taking the "shortest" route between two events (the event where the twins separate and the event where they meet again).

If we solve the Euler-Lagrange equation for the spacetime interval case, we again get $x=At + B$ as the extremizer, but this time it acts as a maximizer (because of the Lorentzian signature of the spacetime pseudo metric tensor). A geodesic in flat spacetime is just constant velocity motion, so the spacetime interval for a geodesic must be maximum (and yes, I know this is true for non-flat spacetime geodesics too [we can derive the geodesic equation from the concept of maximized spacetime intervals]). This brings me to problem number 2.

2)

however, it is easy enough to see that if ds2ds^2 is to be minimum (generalization of Newton's first law), the spatial distance must be minimum ( as expected) but dt2dt^2 must be maximum. Any acceleration is going to result in ds2ds^2 becoming greater than its minimum value, and dt2dt^2 becoming less than its maximum value.

This is just wrong. I don't know what was going through my mind when I typed this, but I've already given the math above to show that what I typed here is wrong. Generalization of Newton's first law will mean that the spacetime interval will be maximum. Since $d\tau ^2 =ds^2$ (again, using the ---+ convention for convenience), proper time must be maximized too. In this sense, the length of the worldline (the line element on the manifold) is the age of the object. Agreed.

@PeterDonis @Orodruin It must have been an eyesore to see such a basic error. Have I made satisfactory reparations in this follow-up post?

P.S. I'm switching from soda to coffee now. Seriously.

 

[PeterDonis]

If we solve the Euler-Lagrange equation for the spacetime diagram "length", we get the function $x=At + B$ which acts as the extremizer, where AA and BB are constants. This function is a minimizer here, so v=v = constant is going to result in the "shortest" worldline on a spacetime diagram.

This is still wrong, because you're assuming that minimizing the Euclidean length is equivalent to maximizing the spacetime length. That is not true in general; it happens to be true for the particular case that you picked, but other cases have been given in this thread where it is not.

When we say the Euclidean length has no physical significance, we really mean "no". You can't use it for anything. You shouldn't even mention it at all. It just causes confusion if you do.

Have I made satisfactory reparations in this follow-up post?

Not entirely. See above. What you say about the spacetime length is fine, though

I'm switching from soda to coffee now.

IIRC, Mountain Dew has as much caffeine as coffee, so if you're really partial to soda, you still have an option.

 

[MarcusAgrippa]

Hmm so when I fly to Alpha Centauri at 0.9c, I only age 1.83 years.
How do you explain it without length contraction?
How can you even use Lorentz transformation without referring to length contraction?

Use proper time. This requires neither a Lorentz transformation, nor any calculation involving length contraction or time dilation.

Proper time is the time read by an on-board clock - the number of ticks of the clock, if you like. We all have an on-board clock, a ticker that one day will stop, short, never to go again. Our on-board clock is not an ideal clock - occasionally it skips a beat, or races when we see a pretty girl or hunk of a man, but the number of times it ticks determines our age. This ticker, over a long time, will tick on average at the same rate as an ideal clock - well, roughly so.

Proper time along two different world lines that begin at the same event A and end at the same event B display different proper times elapsed from A to B. The persons whose world lines they represent thus age by a different amounts. The longest proper time lapse is along the straight world line joining A to B. This is easy to show by a simple geometric argument requiring no calculation. The time lapse along any other curved (timelike) world line joining A to B is shorter, and so the person whose world line this is ages less between events A and B. This offers a simple and complete resolution of the twin paradox, without any computations, by simply understanding the physical significance of proper time.

 

[Buzz Bloom]

All they tell you is some unobservable weirdness associated with two moving objects

Hi m4r35n357:

I am a bit puzzled by your use of the word "unobservable" as applied to time dilation in SR. I am thinking of a particle of a type with a known measured half-life "living" much longer than its known (non-relativistic) half-life when traveling (relative to an observer) at relativistic speeds.

Regards,
Buzz

 

[bligh]

Great reading above!
As one of the "newbies" more intermediate in understanding, I would say. Where is there something to read that sums up consensus in early 21st century?
My incomplete knowledge is this.
1 absolute time and space are real but unobtainable. Therefore I suggest a locally accurate Universal Reference Point, as a point midway between 3 local galactic centers. It is held in place automatically at equal distances from the 3 gal. centers. The URP would be theoretical but its attachment to abs space and time is that locally it would be a proxy for "absolute" frame of reference. It would determine what is actually happening in terms of distance and velocity.
My understanding is that Time dilation is accepted as shown by GPS. My understanding is that a cesium clock on the GPS would tick at a slower rate physically relative to a URP. My understanding is that length contraction has never been demonstrated without good argument. As far as mass getting larger with increased velocity that is an idea related to inertia, and inertia is only acceleration relative to the underlying field that produces mass in the first place. And humans age based upon their own personal history which is based upon the Ref Frame they biologically evolved in. Proof is in physiological changes observed in our own "space" travelers on the International Space Station.
I am not a crackpot. If I misunderstand I would like a reference or two to set me straight. I is obvious to me that my observations may or may not be incorrect, judging from the preceding comments.
Bligh

 

[Alan McIntire]

I didn’t see any mention of Doppler shift here, so I’ll give my 2 cents worth based on a book I read by PJE Peebles.

Assume Alpha Centari is exactly 4 light years away, and one twin is
traveling
there at 4/5 speed of light. (Using a 3,4,5 triangle I avoid
irrational numbers in my computations.

Traveling at 4/5 the speed of light, from the point of view of the
stay at home twin, the trip will take 10 years, 5 years there, 5 years
back.

Time for the traveller T’ = T( sqrt( 1- (v^2/c^2))) = 3/5 T
Likewise, the distance for the traveler, D’ = 3/5 D

The traveler on the spaceship sees himself traveling a distance of
4*3/5 = 2 2/5 light years in a time of 3 years, and likewise the 2 2/5
light years back
in a time of 3 years, so the traveler will see the trip as lasting 6
years.

Say the twins have super telescopes and can see each other throughout
the trip.
As long as they are traveling apart, the twins will see each other as
aging at 1/3 speed. As long as they are traveling towards each other,
the twins will see each other as aging at triple speed.

The difference is, the traveling twin will see the stay at home twin
as aging at 1/3 speed for the 3 years to Alpha Centauri, for a total
of 1 year,
and at triple speed for the 3 year trip back to Earth = 3*3=9.
The traveling twin will see the stay at home age 1 year during the
trip out, and 9 years during the trip back, for a total of 10 years.

The stay at home twin will see the travel age at 1/3 speed for 9
years, the 5 years it takes the traveler to get to Alpha Centauri,
plus the 4 years it takes the light to get back to earth. Since the
total trip will take 10 years, the stay at home twin will see the
traveler age at triple speed during the one year he observes the
traveler coming back to earth. The Earth observer sees the traveler
age at 1/3 speed for 9 years, for a total of 3 years, and at triple
speed for 1 year, for another 3 years, giving 6 years for the round
trip.

Both observers see each other aging at the same slow rate while moving
apart, they see each other aging at the same fast rate while moving
together. The difference lies in one observer deliberately changes
the relative motion of his rocket from moving away from Earth to
moving towards earth, and the other observer remaining passive, and
not seeing the change until the light from the
traveler reaches earth. If the Earth could be accelerated like a
rocket ship, and the earthbound observer decided to change his frame
so the rocket appeared to be moving towards him at 4/5 lightspeed
rather that away at 4/5 lightspeed, while the rocket remained in
motion past Alpha Centauri, then it would have been the Earth twin who
appeared to age less.
Of course you could have some intermeditate situation where BOTH
observers decide to change their relative motion before they see the
other observer change his motion.

We get our prejudices on absolute time from living on earth, where communication is within 1/10,000 of a second. We cannot see stars or planets "now". we see them
light years away for stars, light minutes or hours away for planets.

 

[PWiz]

That is not true in general;

But I'm not talking about the general case; I'm only considering timelike trajectories in flat spacetime. Then minimizing the Euclidean distance (on the spacetime diagram) automatically maximizes the spacetime interval.

IIRC, Mountain Dew has as much caffeine as coffee, so if you're really partial to soda, you still have an option.

A tempting alternative indeed.

 

[PeterDonis]

I'm not talking about the general case; I'm only considering timelike trajectories in flat spacetime. Then minimizing the Euclidean distance (on the spacetime diagram) automatically maximizes the spacetime interval.

If you're only talking about one single timelike geodesic, yes, this is true. But the discussion in this thread is considering a wider scope than that; it is considering "triangles" in spacetime composed of three timelike geodesic sides. For that case, the rule breaks down. That being so, I don't think talking about the rule in Euclidean terms has any value, since questions along these lines always come up when considering "triangle" type scenarios like the twin paradox.

 

[SlowThinker]

As long as they are traveling apart, the twins will see each other as
aging at 1/3 speed. As long as they are traveling towards each other,
the twins will see each other as aging at triple speed.

Where do these numbers come from? The easiest way to find them seems to be working backward from the final result obtained in some other way.

Also, this thread isn't really about the Twin Paradox and the various ways to arrive at the result. We're debating if there is a scenario where the use of time dilation is useful. If you wanted to show a simple way to arrive at $\sqrt{(4/0.8)^2-4^2}=\sqrt{(4/4*5)^2-4^2}=3$, I'm afraid you failed.

 

[PAllen]

Where do these numbers come from? The easiest way to find them seems to be working backward from the final result obtained in some other way.

Also, this thread isn't really about the Twin Paradox and the various ways to arrive at the result. We're debating if there is a scenario where the use of time dilation is useful. If you wanted to show a simple way to arrive at $\sqrt{(4/0.8)^2-4^2}=\sqrt{(4/4*5)^2-4^2}=3$, I'm afraid you failed.

The standard formula for collinear relativistic Doppler factor is:

√((1+β)/(1-β)) for approach
and
√((1-β)/(1+β)) for separation

 

[polydigm]

First a question. Is everyone here setting c=1 in their calculations? I was taught ds²=c²dt²-dx², with ct as time axis in Minkowski space.

On the discussion earlier about rates of ageing, I agree with the notion that these are not different in different frames. The twin leaving at age 20 and travelling, to return at age 30 to find his sibling at age 50, will not have been able to achieve 30 years of effort while he is away. He will not have aged more slowly than his sibling. He will have traveled for 10 years in his own frame and it will have seemed like 10 years when he returns. So, it is definitely not about rates of ageing.

Medical science has effectively slowed rates of ageing over the last century or so. People live longer lives. Relativity will not enable you to live a longer life, so different rates of ageing is not a useful concept.

 

[Dale]

And what term other than "time dilation" would you propose to describe the fact that one person has aged at a different rate than the other or some physical process has progressed at a different rate?

I have thought a bit about this and (4 pages later) I am inclined to agree.

One reason to avoid the term "relativistic mass" is that we already have the term "total energy" which describes the same thing. As you point out, that is not true here. There is not another name referring to the same quantity as time dilation.

Certainly you could prepare your curriculum without overemphasizing it, but it seems to be at least a non-reduntant term.

 

[PeterDonis]

There is not another name referring to the same quantity as time dilation.

But "time dilation" is used to name two different concepts: the frame-dependent concept of "rates of time flow" being different from one frame to another, and the frame-independent concept of two different timelike worldlines between the same two events having different lengths. So to avoid ambiguity, one of those concepts needs to have a different name. That's why I proposed "differential aging" for the second one.

 

[Dale]

So to avoid ambiguity, one of those concepts needs to have a different name. That's why I proposed "differential aging" for the second one.

Yes, I think that is the better approach.

 

[Mister T]

But "time dilation" is used to name two different concepts: the frame-dependent concept of "rates of time flow" being different from one frame to another, and the frame-independent concept of two different timelike worldlines between the same two events having different lengths. So to avoid ambiguity, one of those concepts needs to have a different name. That's why I proposed "differential aging" for the second one.

Is it really that simple, though? For the traveling twin a proper time $\Delta \tau$ elapses during his entire journey. The stay-at-home twin calculates that the time $\gamma \Delta \tau$ will elapse on his clock. Is this a dilated time, differential aging, or both?

 

[Dale]

Is it really that simple, though? For the traveling twin a proper time $\Delta \tau$ elapses during his entire journey. The stay-at-home twin calculates that the time $\gamma \Delta \tau$ will elapse on his clock. Is this a dilated time, differential aging, or both?

That is invariant so it is differential aging.

 

[PAllen]

Yes, I think that is the better approach.

Differential aging is a relatively standard term. Not sure where/when I first saw it, but it's a term I've used as distinct from time dilation for many years. One is invariant and one is frame dependent. Note that frame dependence doesn't necessarily mean unobservable, because you can materialize a frame implementing standard clock sync and observe standard clock synch between separated clocks. Any observer, analyzing your set up, would correctly predict your observation.

Note also that muons reaching the ground is an observation, and is not differential aging. Being a non-local series of observations (you have to find out that muons are created high in the atmosphere), it has multiple frame dependent explanations - time dilation or length contraction.

Based on this, I'm not so sure these concepts should be so deprecated just because they are frame dependent. Yes, you can just do an interval computation in any coordinates, but it seems useful to me to have terms to describe this scenario.

 

[Dale]

Several off topic posts have been removed

 

[Mister T]

That is invariant so it is differential aging.

So, is this an example of the dilated time $\gamma \Delta \tau$ being invariant? Or is it an example of $\gamma \Delta \tau$ not being a dilated time?

Or is there a third option I'm not seeing?

The thing that makes $\gamma \Delta \tau$ invariant in this case is that it's a proper time (time between two events that are not spatially separated) for the stay-at-home twin. So this makes it a special case. The thing I want to know is when I teach the twin paradox to non-majors am I cheating when I call this time dilation?

 

[PeterDonis]

The thing I want to know is when I teach the twin paradox to non-majors am I cheating when I call this time dilation?

I don't know that it's "cheating", but it might be confusing, since, as I said in a previous post, the term "time dilation" has two possible meanings. One is the invariant thing you describe. The other is something that is not invariant; it's frame-dependent (the fact that a moving clock "appears to run slow", which depends on your choice of frame). Using the same term for two things, one of which is invariant and one of which isn't, is going to cause confusion. As far as I know, nobody has tried any term except "time dilation" for the frame-dependent thing, so that seems like the best one to keep the term; but then we need to find a different term for the invariant.

 

[m4r35n357]

I don't know that it's "cheating", but it might be confusing, since, as I said in a previous post, the term "time dilation" has two possible meanings. One is the invariant thing you describe. The other is something that is not invariant; it's frame-dependent (the fact that a moving clock "appears to run slow", which depends on your choice of frame). Using the same term for two things, one of which is invariant and one of which isn't, is going to cause confusion. As far as I know, nobody has tried any term except "time dilation" for the frame-dependent thing, so that seems like the best one to keep the term; but then we need to find a different term for the invariant.

BTW thanks to everyone who has commented on this thread, I've enjoyed it more than I thought I would (I suppose I expected to be slagged off or ignored)!

I'm probably going to show my ignorance here, but that will only serve my point that the concepts are confusing ;) I most commonly hear/read about TD & LC in the context of just multiplying/dividing the two "observer" frame quantities by gamma. Now, if I have understood the Lorentz transform correctly, the "moving frame" length needs to be measured at different times, and the "moving frame" time at different positions. This is perhaps the root of my feeling that they are both unobservable, and rather contrived.

Now, since gamma is encoded in a rather obvious way in the spacetime interval, I am not surprised that people use the "divided" quantities as if they are nothing unusual, but I do find with that usage clumsy and "untidy".

Also, the idea that the rest-frame distance obligingly compresses itself at the whim of a traveler seems misleading too. I maintain that the (coordinate) distance traveled is just that (I suppose you can say that it "stretches back" when you get there, but, well, bah!). The time dilation is not so obviously misleading, but really what is wrong with "proper time" ? Aging is another word for that, but even that sounds a bit anthropomorphic to me.

So, now I have laid bare my (mis)understanding of the two concepts, perhaps you can understand why I find them confusing, and avoid them with a vengeance! When using the spacetime interval I can at least pretend to know what I am talking about ;)

I shall now brace myself . . .

 

[Alan McIntire]

In reply to slow thinker, the OBSERVED doppler effect is sqrt {(1-v/c/1 + v/c)} for objects moving apart,
sqrt {(1 + v/c)/(1-v/c)} for objects moving together. Plug in 4/5 c for v and you'll get 1/3 and three respectively

 

[PAllen]

In reply to slow thinker, the OBSERVED doppler effect is sqrt {(1-v/c/1 + v/c)} for objects moving apart,
sqrt {(1 + v/c)/(1-v/c)} for objects moving together. Plug in 4/5 c for v and you'll get 1/3 and three respectively

see post #84.

 

[SlowThinker]

I most commonly hear/read about TD & LC in the context of just multiplying/dividing the two "observer" frame quantities by gamma. Now, if I have understood the Lorentz transform correctly, the "moving frame" length needs to be measured at different times, and the "moving frame" time at different positions. This is perhaps the root of my feeling that they are both unobservable, and rather contrived.

Well I've been reading and thinking about TD&LC for so long that it feels natural that as my spaceship accelerates towards Alpha Centauri, the distance gets shorter and I age less during travel.
The main trouble I see is that the Relativity of Simultaneity is the most important effect, that
- can be explained without the need to understand TD&LC
- RoS is necessary for TD&LC to make sense
There are 2 sides to RoS:
When talking about a length-contracted spaceship, you always have to remember that the clock in the front show different time from the ones at the back (I'm always having trouble remembering which is which, probably later/older at the front).
Also, as the spaceship travels, it arrives at places where time is running slow, but there *already is* the future. So you still arrive at the destination 4 years later even if you spent just 1 year flying.

Hope this helps... or just ignore me

 

[SlowThinker]

In reply to slow thinker, the OBSERVED doppler effect is $\sqrt {(1-v/c)/(1 + v/c)}$ for objects moving apart, $\sqrt {(1 + v/c)/(1-v/c)}$ for objects moving together. Plug in 4/5 c for v and you'll get 1/3 and three respectively

Yes but that's introducing yet another effect, more derivations, and more formulas to remember.

 

[robphy]

In reply to slow thinker, the OBSERVED doppler effect is sqrt {(1-v/c/1 + v/c)} for objects moving apart,
sqrt {(1 + v/c)/(1-v/c)} for objects moving together. Plug in 4/5 c for v and you'll get 1/3 and three respectively

Yes but that's introducing yet another effect, more derivations, and more formulas to remember.

In Bondi's method, the doppler effect is more primitive (the Doppler Factor is an eigenvalue of the Lorentz Transformation)...
other effects (like time-dilation, length-contraction, velocity-composition, and the Lorentz transformation) are then derived from it.

 

[m4r35n357]

Yes but that's introducing yet another effect, more derivations, and more formulas to remember.

Hold on a second, I would definitely advocate learning about the Doppler effect. It is VERY real and essential to almost all astronomical and cosmological studies (and makes analyzing the twin paradox a doddle). Also the aberration of light (see steps 3 & 4 of my OP!). This is exactly the sort of interesting stuff that you can get into once you get over the trivia ;)

 

[SlowThinker]

Hold on a second, I would definitely advocate learning about the Doppler effect. It is VERY real and essential to almost all astronomical and cosmological studies (and makes analyzing the twin paradox a doddle). Also the aberration of light (see steps 3 & 4 of my OP!). This is exactly the sort of interesting stuff that you can get into once you get over the trivia ;)

Oh, I thought that the Doppler effect is used *in addition* to Lorentz transformation, not *instead of* it. It makes sense then, although I'm not really used to thinking in terms of Doppler effect.

Are you saying that it's worth it learning about aberration (I agree), or that the Doppler effect explains it? (How??)

 

[m4r35n357]

Oh, I thought that the Doppler effect is used *in addition* to Lorentz transformation, not *instead of* it. It makes sense then, although I'm not really used to thinking in terms of Doppler effect.

Are you saying that it's worth it learning about aberration (I agree), or that the Doppler effect explains it? (How??)

The Doppler effect and aberration both have the LT "built in". The Doppler effect shows how colours of objects change with relative motion, whilst aberration deals with changes in the position/shape of objects. Both these effects are visual, so you need 2 space dimensions and one time to describe them.

Please take a look that these videos I made, and read the explanation. There is a lot to take in, so don't expect to understand what is going on straight away, but treat it as a pointer to where SR can take you if you follow my advice (sorry if this sounds pretentious, it's not my intention).

BTW just to clarify, I might have given the impression that I oppose over-stressing the importance of TD/LC out of laziness. This is most definitely not the case, I do it in the interests of efficiency.

 

[SlowThinker]

whilst aberration deals with changes in the position/shape of objects. Both these effects are visual, so you need 2 space dimensions and one time to describe them.

So you replace the length contraction with aberration?

Please take a look that these videos I made, and read the explanation. There is a lot to take in, so don't expect to understand what is going on straight away, but treat it as a pointer to where SR can take you if you follow my advice (sorry if this sounds pretentious, it's not my intention).

I've seen the videos a few days ago but I can only access the first line of the explanation, so I can't quite understand what's going on.

 

[m4r35n357]

So you replace the length contraction with aberration?

I've seen the videos a few days ago but I can only access the first line of the explanation, so I can't quite understand what's going on.

See the bit where it says "more"? It's bedtime in the UK so I'll let you get on with it for now . . .