Is the Changing Clock Rate in Relativity Directionally Dependent?

[yogi]

robphy - I agree that a Nobel award does not qualify a writer as an authority - i guess i wanted to get across the idea that there is disagreement among respected authors - persons who are presumably a cut above the cranks.

I once made a list of statements made by recogonized authors re relativity - in particular those that dealt with the reality of spatial contraction and time dilation ...like the bible, there was support for every theology.

 

[Ich]

sorry, I´m quite out of sync, but maybe it´s worth to work out my point.
yogi - i didn´t make myself very clear in post #26. The analogy is as follows:
Two cars with the same speed - two observers with the same four-velocity (c or 1, whatever you prefer).
Two cars heading in different directions - two observers in relative motion.
The core of the analogy is: each car will symmetrically see the other fall behind. That´s what you would call "apparent". But if they match directions, one of those cars will really be behind the other. That´s what you would call an "actual difference".
Translate this to Einstein´s chapter 4: car A initially drives parallel to car B. Then it steers towards car B until it is on the same track. Of course it will be behind car B in this track. And of course you wouldn´t bother about "real" and "apparent" car positions. You wouldn´t even claim that car A is slower than B in an absolute sense while it steers towards B.

 

[RandallB]

the cause of actual time dilation

Again; in classical SR there is no “actual time dilation”, no clock ever “actually” runs fast or slow, thus there can be no time slippage for a individual clock.

So when you say you have a simple answer. But when I go through the reasoning - always find a transition - a shift from observation to reality - non-linear system - somewhere there is jump from apparent clock rates to actual time slippage.

Not my solution or answer – just a complete workup of the Classical math applied to the SR factors.

Look at the rates and ratios SR uses, once your committed to selected fixed speeds there are no “non-linear” functions. It’s all algebra, you cannot have a conflict in measurement between reference frames or transition from one observation to another’s, algebra doesn’t work that way. Only errors in the math, so double check the work, not by just doing it again but by completely solving for the same numbers from within each of the 3 reference frames.
Even if some “clocks” need to run fast!
Example, as the observer sits in the station waiting for the twin to return as the train goes by clocks in each car can be seen flashing creating an animated clock like one of those cartoons drawn in a stack of cards you flip through. Isn’t that also in the traveling twins reference frame where all those clocks are synchronized with each other. It should be just as valid for defining the age of the traveling twin. What formula would you use to define this clock time based on the station clock time. If it doesn’t match with your expectation of the clock traveling with the twin, when is the “imaginary” clock the correct way to gage the age of the traveling twin??
Don’t conceptualize on it, don’t look to me or anyone for an opinion, just do all the Classical SR algebra for all the variations yourself. Then see if you cannot absorb your own work till you “get it”.

- but instead I will probably just go skiing

Don’t give up on yourself, you can do this algebra in an evening and absorb it in a weekend.

 

[yogi]

RandallB - its not that I won't do the algebra - I have done it many times from different perspectives - one telephone call to my wife and you would be convinced that I fill up pad after pad of paper with equations and notes almost every night - I go through the derivations until I can not think anymore - then hit the sack - generally about 3 or 4 or the morning - she gets up and finds a mess of scribbling. Maybe that's the problem - too late - too many beers

 

[yogi]

ich - i want to consider your post 37 more before responding

Thanks

yogi

 

[yogi]

Ich - I must be missing something - when Car A and B are moving parallel at the same speed the relative velocity is zero - no time dilation. When car B turns and drives toward A there is a relative velocity v and each would judge the time in the other car to be running slow - there is apparent time dilation in this case

 

[RandallB]

RandallB - its not that I won't do the algebra - I have done it many times from different perspectives - one telephone call to my wife and you would be convinced that I fill up pad after pad of paper with equations and notes almost every night - I go through the derivations until I can not think anymore - then hit the sack - generally about 3 or 4 or the morning - she gets up and finds a mess of scribbling. Maybe that's the problem - too late - too many beers

beerS - Maybe so,
Think of it like a check book register. You can keep the total balance several different ways. A running total balance as each record is entered; A running balance of deposits separate from checks, only combining the two when you need to now a current total.; Or only summing deposit and check balances by page & only summing the page totals when you need to work a total balance. Three different way to get exactly the same result, that can all be done at the same time serving as a triple check of the balance because if anyone disagrees there must be a math error somewhere.

Working SR is the same way. I assume you start with three reference frames, two going the same speed but in opposite directions of a third (usually assigned to earth).
And define a time = 0 and location = 0 to be local to all three frames at that one instant in time.
Now for any location at any time you pick in any frame it will have only one location in each of the other two frames local to it. And each of those will have their own
time that can be “seen” (calculated) from the frame you’re working from.
Now the point is when you redo all those measurements from any other frame you must come up with the same combinations of times and locations local to each other for any giving point and time in a frame.
Such as location #0 (Station, train car, space ship, whatever based on + or – distance) at a different time say +20 or -20 time units, there is one and only one location, with its own unique time, in each of the other two frames that is local to that point and time.
But you can double and even triple check those numbers by also calculating them completely from within those other frames.

If while doing those cross checks you find any of those sets of six numbers for an individual location and time, not be identical with the other calculations. Then there can only be one of two problems:

1) There is an error in the math or its application; it’s just a matter of finding it, just like balancing a check book.
OR
2) Unlike a check book there is something about the lambda and dilation factors being used that are not linear; this leading to a condition called background independence that will make it impossible to “balance” these kinds of cross checks; BUT you should be able to define where that non-linear part of the formulas is causing the problem.

But where in SR is there anything non-linear? Sure GR starts into curving and warping things and is accepted by many, even most, as being background independent. That’s a problem for GR not SR. I think IMO the problems you find in most books and these posts on SR are too short, vague, or incomplete (including Einstein’s) to make the point clear.

You’ll just have to decide for yourself; is it #1 and fix it;
or #2 and find the non-liner piece.

Try A Barleywine instead of beers; maybe that will help.
I find Bigfoot aged two years in bottle to be excellent, but fresh is very good too.

 

[Janus]

Ich - I must be missing something - when Car A and B are moving parallel at the same speed the relative velocity is zero - no time dilation. When car B turns and drives toward A there is a relative velocity v and each would judge the time in the other car to be running slow - there is apparent time dilation in this case

I think you are missing the point of the analogy.
Each car respresents a clock.
The path each car travels represents progression through time as determined by that clock.
The angle of the two cars' paths to each other represent the relative velocity between the cars.

Each clock(car) measures the other clock(car)'s progression through time as compared to its own present heading.(draw a line that run through the clock(car) along the line that the clock(car)1 is pointed. Now draw a line perpendicular to this line that intersecrts the other clock(car)2. Where this line crosses the first line, is how much time clock(car)1 will determine has passed for clock(car)2.

Clock(car)2 will determine clock(car)1's progression through time by the same procedure.

A change in velocity is represented by a change in heading.

If clock(car)2 changes its velocity to match clock(car)1, this shown by its changing its course until it parallels that of clock(car)1. After which clock(car)1 and clock(car)2 both measure each other as progressing through time at the same rate, and clock(car)1 will be ahead of clock(car)2 (has accumlated more time) according to both cars.

 

[RandallB]

Ich & Janus
I really don’t see how making the problem more complex by introducing vectors of undefined angles makes anything any easier.
There is nothing that can do that isn’t more simply represented by different speeds defined for reference frames moving along one straight line.

The only thing needed are different fixed speeds and instant transfers between frames without accelerations of GR to account for. (Transfer time 0 as seen by all frames)

Why make it more complex than that?

 

[Ich]

RandallB,
I did not intend to make things more complex; instead I tried to translate yogi´s problem of real and apparent effects to an example of every day life.
To me, the SR question "which clock really ticks slower" is absolutely equivalent to the non-SR question "which car is ahead" when they drive in different directions. It simply does not make sense.
Yogi´s conceptual problem of "real" vs "apparent" arises in a nearly perfect analogy in our every day experience. But it´s resolution is much easier when you consider the example of the two cars.
Of course, first of all it´s most important to get the point of the analogy. Janus kindly explained it some more where I obviously failed.

 

[RandallB]

Ich
OK, but it still retains some of the implication that maybe some clock does "really tick slower" which is not true.

That’s why is important to build an example that reveals the full nature of simultaneity to show that no clock ever runs fast or slow.

 

[Hurkyl]

Since it's not a technical term with a clear definition, I spent some time earlier today trying to figure out what you might mean by words like "illusory".

The first two situations that popped in my mind were these:

If I'm in a car driving towards the source of a sound, I will hear it at a higher pitch. I could imagine someone calling this illusory.

If I'm in a car, I see person sitting next to me in a fixed place. I could imagine someone saying that I see that person as stationary was illusory.


Both observations are certainly real. Although the musician was trying to play a 440 Hz note, it really did triger that 460 Hz region of my ear. And when I hold out my ruler to measure your position, the passenger is always right there at the end.


These two examples have a common element: that there is a commonly agreed "right" way to make these measurements. E.G. it is the common convention that we measure speeds relative to the ground, as opposed to ourselves.

So, in these examples, the usage of "illusory" would simply mean that the measurements were performed in reference frames different from the commonly agreed one.


This usage of "illusory" is fundamentally different from other situations, such as that of a magician who hides things behind curtains, or a demon who sends you misleading light signals, all to deceive your vision.


If I recall correctly, your usage of "illsuory", yogi, is akin to the first two examples I stated.

Usage of this term in such a circumstance is highly misleading (and I think you have even misled yourself), since the word carries the negative connotation of the magician or demon who is actually tricking you.

 

[yogi]

A lot to answer - but as far as illusory - it has to me the same meaning as a measurement of distances in a relatively moving frame - if there is a real measurment of a 10 foot pole that appears to be 8 feet - I would call that illusory - Hurkyl - some time back we had a private discussion and I responded to your explanation that the one twin is halfway there because of the "time jump" before he starts - and I drew an analogy to the explanation offered by Rindler which gets the same result - namely that the traveling twin travels a shorter distance to Vega (or whatever star you pick) and therefore less time is involved because the distance is half to start with - I know you get the desired answer - but the method subverts the issue - it makes the rationale more important than the reality principle ...arriving at the right numerical value for the wrong reason does not help if the object is to understand the underlying physics.

In summary - the fact that the traveler sees the distance from Earth to Vega is not a bases for time dilation - its the other way around - time dilation is the cause - the consequence is that the traveler measure the distance to the object to be shorter because it is the reality of temporal changes that is fundamental - not vice versa.

let me try to get back after the sideline discourse on what I meant by illusory. Here is a situation from the real world - we have two clocks A and B both at rest at a 100 mile high tower at the North Pole - then we put A in orbit in a GPS satellite at the same altitude - but prior to launch we set it to run faster to compensate for the orbital velocity.

Each time A pases over the North Pole, the orbiting clock A will be in sych with the Earth clock B- but let's say we forget to make the velocity correction before launch - the orbiting clock A will continue to lose time on each pass - this is not a case where each clock sees the other to be running slow - the tower clock B sees the A clock running slow, and the A clock sees the tower clock B to be running fast. This is real time dilation - it is an intrinsic result of the fact that they were initially synchronized at the top of the tower and A is accelerated into orbit. It has nothing to do with the fact that the orbiting clock A is moving in a circle - A is in an inertial free fall environment. Isn't this a straight forward verification of Part 4 of Einstein's paper?

 

[yogi]

Now - while A is in clockwise orbit we arrange to build another clock J on board and sync it with A; each pass by the tower, A and J are seen to fall progressively behind B. Now J is accelerated in a counterclockwise direction until it comes to rest on the tower. Because A and J were initially synchronized while they were traveling together, the reverse acceleration of J away from A presents an interesting situation when analysed from A's frame, but not from the tower frame. From the Tower frame, it makes no difference whether A or J returns to B since both will have been running slow wrt to B by the same amount, either will run at the B rate upon arrival at the tower.

 

[Hurkyl]

if there is a real measurment of a 10 foot pole that appears to be 8 feet - I would call that illusory - Hurkyl

Illusory or not, you can still fit it through a 9 foot barn door the long way. (As measured from Earth's inertial frame)


Allow me to explain this one in more detail:

I will always carry the 10-foot pole so that it is pointing in the East-West direction. (As measured in the Earth frame)

There is a 9-foot wide door on the South side of the barn.

By running sufficiently fast in a Northeasterly (or Northwesterly) direciton, I am able to take the pole inside the barn.

arriving at the right numerical value for the wrong reason does not help if the object is to understand the underlying physics.

Why do you think it's the wrong reason? It almost seems that you're of the opinion that there can only be one way to solve any problem!

and the A clock sees the tower clock B to be running fast.

Wrong. Clock A will observe clock B to be running slow, when they're near each other.

 

[yogi]

Hurkyl - if the tower clock B reads one hour on the first pass, 2 hours on the second, 3 hours on the third etc, in other words a precise 60 minutes per orbit, and the orbiting clock A reads 59 minues, 118 minutes, 177 minutes etc on each successive fly-by, the orbiting observer with clock A will note that B is gaining one minute each orbit, ergo, B will conclude A is running fast wrt to his own measurement of time. This is an example of real time dilation - failure to correct for the relativistic velocity of A prior to launch will cause the two clocks to accumulate different times during each successive orbit.

 

[JesseM]

Each time A pases over the North Pole, the orbiting clock A will be in sych with the Earth clock B- but let's say we forget to make the velocity correction before launch - the orbiting clock A will continue to lose time on each pass - this is not a case where each clock sees the other to be running slow - the tower clock B sees the A clock running slow, and the A clock sees the tower clock B to be running fast. This is real time dilation - it is an intrinsic result of the fact that they were initially synchronized at the top of the tower and A is accelerated into orbit.

It has nothing to do with the fact that A rather than B was the one that was accelerated after they were initially comoving, if that's what you mean. If instead A and B were two synchronized clocks orbiting together, and then B was accelerated so it came to rest on top of the tower while A continued on its freefall path, the time dilation effects on subsequent orbits would be exactly the same.

It has nothing to do with the fact that the orbiting clock A is moving in a circle - A is in an inertial free fall environment. Isn't this a straight forward verification of Part 4 of Einstein's paper?

Well, if you tried to analyze this problem using only SR you'd have to ignore the curvature of spacetime caused by the Earth's gravity, in which case A would not be moving inertially. Even in the context of GR, I'm not sure if it would be correct to say that A is moving "inertially" even though it is in free fall, I don't know if the concept of inertial vs. non-inertial motion makes sense except in a purely local sense (in an arbitrarily small region of spacetime which is arbitrarily close to flat) in GR. In any case, it definitely makes no sense to say that a situation which involves curved spacetime in an essential way is a "straight forward verification of part 4 of Einstein's paper" when that section dealt purely with velocity-based time dilation in the flat spacetime of SR.

 

[Hurkyl]

Hurkyl - if the tower clock B reads one hour on the first pass, 2 hours on the second, 3 hours on the third etc, in other words a precise 60 minutes per orbit, and the orbiting clock A reads 59 minues, 118 minutes, 177 minutes etc on each successive fly-by, the orbiting observer with clock A will note that B is gaining one minute each orbit, ergo, B will conclude A is running fast wrt to his own measurement of time. This is an example of real time dilation - failure to correct for the relativistic velocity of A prior to launch will cause the two clocks to accumulate different times during each successive orbit.

You're only looking at the "average".

Everybody will agree that, over the long term, the tower clock runs faster than the orbiting clock, for the reasons you describe.

I think that it would be inaccurate to call this "dilation", though, since you're considering a discrete series of events.


However, as the tower and orbiting clock pass each other, the orbiting clock will observe the tower clock running slowly. (and vice versa)

I don't know if the concept of inertial vs. non-inertial motion makes sense except in a purely local sense (in an arbitrarily small region of spacetime which is arbitrarily close to flat) in GR.

I thought that "inertial" was interpreted to mean "travelling along a geodesic"?

 

[JesseM]

I thought that "inertial" was interpreted to mean "travelling along a geodesic"?

Is an object in freefall said to be moving "inertially"? Have you seen specific examples (in textbooks, say) of physicists using "inertial" in this way in the context of GR?

 

[Hurkyl]

I don't remember.

 

[robphy]

I thought that "inertial" was interpreted to mean "travelling along a geodesic"?

Is an object in freefall said to be moving "inertially"? Have you seen specific examples (in textbooks, say) of physicists using "inertial" in this way in the context of GR?

In GR, an object in freefall is said to be moving "inertially", "geodesically".

 

[JesseM]

OK, I'll take you guys' word for it. Still, it seems to me that yogi's statement "It has nothing to do with the fact that the orbiting clock A is moving in a circle - A is in an inertial free fall environment" is confusing the SR meaning of "inertial" and the GR meaning of "inertial".

By the way, if we calculated the time difference between the two clocks by treating A as if it was just moving in a circle in flat spacetime, does anyone have a sense of how far off this would be from the correct GR calculation in which A is moving on a geodesic in curved spacetime? Would it be close since Earth's gravity is not too strong, or would it be far off?

 

[pervect]

OK, I'll take you guys' word for it. Still, it seems to me that yogi's statement "It has nothing to do with the fact that the orbiting clock A is moving in a circle - A is in an inertial free fall environment" is confusing the SR meaning of "inertial" and the GR meaning of "inertial".

I agree - the clock in a free-fall orbit certainly has a local inertial frame, but it's limited in extent.

By the way, if we calculated the time difference between the two clocks by treating A as if it was just moving in a circle in flat spacetime, does anyone have a sense of how far off this would be from the correct GR calculation in which A is moving on a geodesic in curved spacetime? Would it be close since Earth's gravity is not too strong, or would it be far off?

It will be very close, because $g_{tt}$ is close to 1, and $\sqrt{g_{\phi\phi}}$ is exactly equal to r in the Schwarzschild metric.I.e. SR says that
$$d\tau = \int \sqrt{1- r^2 \, (\frac{d\phi}{dt})^2} \, dt$$

becaause $d\tau^2 = dt^2 - r^2 d\phi^2$, $\phi$ being the angle of the object in its circular orbit.

GR says that

$$d\tau = \int \sqrt{g_{tt} - r^2 \, (\frac{d\phi}{dt})^2} \, dt$$

because $d\tau^2 = g_{tt} dt^2 - g_{\phi\phi} d\phi^2$

here $g_{tt}$ is given by the formula for the Schwarzschild metric e.g.

$$d\tau^2 = (1-\frac{2M}{r})dt^2 - \frac{1}{1-\frac{2M}{r}} dr^2 - r^2 d\theta^2 - r^2 sin^2(\theta)d\phi^2$$

so $g_{tt} = 1-\frac{2M}{r}$

and $\theta$ is zero for an equatorial orbit, so $g_{\phi\phi} = r^2$

(Note: I've used geometric units like I always do, so that c=G=1, adjust for standard units if desired).

 

[yogi]

hurkyl - wrt your post 53 - I agree that if each observer sets up the classical two clock measuring experiment to determine time in the other frame, each could measure the apparent rate of the other clock to be running slow - at least during portions of the orbit - but what I am attempting to say, is that, with reference to the tower clock B, A actually always runs slow during the entire orbit - for example let me construct a plurality of towers equally spaced along on the Earth along the path of the orbiting clock with clocks B through Z all in sync in the Earth centered reference frame. These clocks check the rate of A whenever A passes overhead . Each will read A's clock and see A running slow when it passes near - this is the actual time dilation asserted by Einstein in part 4 - admittedly without a sound foundational bases - but nonetheless verified by experiments that were not conducted until many years later. This is what Einstein is referring to when he concludes "a clock at the equator will run slower than a clock at the North pole"

 

[yogi]

Jesse - as to the orbit as a valid inertial frame - you might check out Spacetime Physics by Wheeler and Taylor - 2nd edition ...they frequently refer to the free fall inertial fame

 

[JesseM]

Jesse - as to the orbit as a valid inertial frame - you might check out Spacetime Physics by Wheeler and Taylor - 2nd edition ...they frequently refer to the free fall inertial fame

I'm still not sure if this means a coordinate system where an object is at rest throughout its entire orbit can be called an "inertial frame", or if this notion of inertial motion only applies "locally", in an arbitrarily small region of spacetime which includes an infinitesimal section of the object's path. This is what I meant when I said:

I'm not sure if it would be correct to say that A is moving "inertially" even though it is in free fall, I don't know if the concept of inertial vs. non-inertial motion makes sense except in a purely local sense (in an arbitrarily small region of spacetime which is arbitrarily close to flat) in GR.

And pervect's last post seemed to suggest it does only make sense to say the path is locally inertial:

I agree - the clock in a free-fall orbit certainly has a local inertial frame, but it's limited in extent.

In any case, aside from the issue of how the word "inertial" is defined by physicists in GR, you didn't address my other points, like the fact that the time dilation effects would be the same regardless of whether A or B accelerated initially, or the fact that an example involving curved spacetime cannot be said to confirm statements made by Einstein in his 1905 paper which dealt purely with velocity-based time dilation in flat spacetime.

 

[yogi]

Jesse - it doesn't make any difference which one accelerates into orbit in the example I have given - If B accelerates into orbit - B's clock runs slower than A's ...but i suspect that is not what you meant ... you are saying things should be symmetrical after one of two synchronized clocks in the same frame is accelerated - and I am saying they are not - and I am also saying that Einstein said they are not. In summary, a literal reading of part 4 is that time dilation in SR is like time dilation in GR - 1) in GR the clock that is at a lower gravitational potential runs slow as measured by the clock at the higher gravitational potential, and the clock at the higher gravitational potential runs fast compared to the clock at the lower gravitational potential and 2) in SR, when one of two synchronized clocks in an inertial system is accelerated to a constant velocity v relative to the other clock, the clock in motion runs slow relative to the clock which was not moved, and the clock which was not moved runs fast wrt to the clock whichn was accelerated.

Only apparent effects are reciprocal (e.g., length contraction). If Einstein says the clock at the equator runs slower than the clock at the pole - he also means the clock at the pole runs faster than the clock at the equator when measured by the reading on the clock at the equator - otherwise they both run at the same speed.

 

[JesseM]

Jesse - it doesn't make any difference which one accelerates into orbit in the example I have given

I didn't say which clock "accelerates into orbit", I said which clock is the one that initially accelerates, period. That was why I offered the example where they both start out traveling together in orbit, then one accelerates until it comes to rest on top of the tower. The clock on the tower will always be the one that runs faster, regardless of whether it's the one that initially accelerated to separate them or not. I was responding to this quote of yours:

This is real time dilation - it is an intrinsic result of the fact that they were initially synchronized at the top of the tower and A is accelerated into orbit.

Weren't you saying here that the time dilation is explained by the fact that the clock in orbit has to accelerate? If so, your point makes no sense, as shown by my alternate scenario where they start out both moving on a geodesic in orbit, then one accelerates (moves on a non-geodesic path) to come to rest on the tower while the other one continues on the geodesic path.

...but i suspect that is not what you meant ... you are saying things should be symmetrical after one of two synchronized clocks in the same frame is accelerated

No, I am obviously not saying that, since that doesn't match the predictions of relativity. See above.

In summary, a literal reading of part 4 is that time dilation in SR is like time dilation in GR

Einstein hadn't even invented GR when he wrote that, so how can you possibly interpret him to be saying anything about GR in part 4?

2) in SR, when one of two synchronized clocks in an inertial system is accelerated to a constant velocity v relative to the other clock, the clock in motion runs slow relative to the clock which was not moved, and the clock which was not moved runs fast wrt to the clock whichn was accelerated.

Are you saying that Einstein or any other mainstream physicist would deny that any situation in SR can equally well be analyzed from any inertial reference frame? Would you deny that in any situation where a clock accelerates and changes velocities, you can find an inertial frame where the clock's final velocity after it finishes accelerating is zero, and that in this frame the clock is therefore running faster after it finishes accelerating?

If Einstein says the clock at the equator runs slower than the clock at the pole

He would only say a clock on the equator runs slower on average over an entire orbit than a clock at the pole. He would certainly never say that a clock at the equator is running slower at every moment, because there are inertial frames where this is not true. But no matter which inertial frame you pick, it will indeed be true that the average ticking rate for a clock moving in a circle will be slower than the average ticking rate for a clock at rest relative to the center of that circle, over the course of an entire orbit.

Every inertial frame is equally valid when analyzing any particular problem in SR. Do you seriously think Einstein would have disagreed with this principle? Note that in any situation where two clocks depart each other and then later reunite, all inertial frames will make the same prediction about which one will be behind when they reunite, so it will do you no good to bring up such situations in an attempt to "disprove" this principle.

 

[pervect]

Every inertial frame is equally valid when analyzing any particular problem in SR. Do you seriously think Einstein would have disagreed with this principle?

Nope. But I don't think he would apply the term "inertial frame" to the orbit of a planet or satellite, either.

Basically, if you are interested in only small distances, if you were on a space-station orbiting the Earth you could totally ignore the tidal forces and variations of the metric, and consider yourself to be in a locally inertial frame, for purposes of walking around inside the space-station. Some very sensitive experiments might still require more careful analysis, but you wouldn't be hit in the face with the non-inertiality of the frame.

But when you start to consider the path of the space station as a whole, the concept breaks down. The distances are to large - the effects of gravity are too large to ignore. They insist on making themselves noticed.

If you actually sit down and write the equations to calculate the Lorentz interval (i.e. proper time), it is very much easier to write down the metric and calculate the proper time from the same POV in GR as it is in SR - with the sun (or other massive body) being at the center. In fact I don't recall ever seeing it done any other way.

If you have multiple orbiting bodies, there is a specific coordinate system that's good for approximate work, too. This approximate coordinate system (used in the PPN formalism) is based on the center of mass of the system. It's not based on an orbiting body. The physics really is simpler with (for example) the Sun as the center of the solar system rather than the Earth.

The orbiting coordinate system may be "locally inertial", but as far as calculations go, it is far simpler to put the origin of your coordinate system at the local center of mass than it is to attempt to deal with the universe "wobbling" around some particular orbit.

 

[Hurkyl]

Each will read A's clock and see A running slow when it passes near - this is the actual time dilation

(from #59)

As A passes through B, C, ..., Z, it will observe each of them as running slowly. It sounds as if you're suggesting that this is not "actual" time dilation -- upon what grounds do you suggest that?

 

[yogi]

Hurkyl - i don't follow what you are saying - in my post 59 I intended to say that the A clock reads the time by looking at the visible counter attached to each tower B, C, D, ...Z the tower clocks always get progressively further ahead. This is not the same as making a measurement using the standard two clock method to determine apparent time dilation in a relatively moving frame - its a simple reading of the counter on the fly.

Now with regard to what has been introduced as an average time loss of each orbit - what I am saying that the orbiting clock A has exactly the same forces, dynamics and whatever else is involved - at every point in the orbit - A clock always runs at its proper time in its Free Float Inertial Frame - the phrase coined by Wheeler and Taylor - nothing changes - therefore what you want to refer too as an average loss of time during one orbit is not average - rather is a sum - you take the time loss in one orbit and divide that by the time for one orbit as measured by anyone of the tower clocks (e.g., B) and you have a real value for the constant rate of time loss as between the two reference frames - The actual rate of passage of time on the A clock does not change during the orbit - this is what I have been referring to as real time dilation -

Jesse - as for your post 63 - i know Einstein didn't invent GR until 10 years later ...I was trying to give you an analogy

 

[JesseM]

Now with regard to what has been introduced as an average time loss of each orbit - what I am saying that the orbiting clock A has exactly the same forces, dynamics and whatever else is involved - at every point in the orbit - A clock always runs at its proper time in its Free Float Inertial Frame - the phrase coined by Wheeler and Taylor - nothing changes - therefore what you want to refer too as an average loss of time during one orbit is not average - rather is a sum - you take the time loss in one orbit and divide that by the time for one orbit as measured by anyone of the tower clocks (e.g., B) and you have a real value for the constant rate of time loss as between the two reference frames - The actual rate of passage of time on the A clock does not change during the orbit - this is what I have been referring to as real time dilation -

You never answer my simple question, which I've asked you a few times:

Every inertial frame is equally valid when analyzing any particular problem in SR. Do you seriously think Einstein would have disagreed with this principle?

Do you agree, incidentally, that if an object is moving in a circle at constant speed in the rest frame of the center of the circle, then its speed will be non-constant in other inertial frames? This is just as true in Newtonian mechanics as it is in relativity, although in Newtonian mechanics of course speed has nothing to do with the rate a clock ticks.

 

[yogi]

Pervect - When we attempt to remove one of two orbiting synchronized clocks to the top of one of the towers - as per jesse's query,do you think it will thereafter run faster or slower than the clock that remained in orbit,
(I know the math is messy - just looking for a conceptual answer if you have one).

 

[yogi]

Jesse as to your post 67- yes - I think Einstein would have disagreed with that - I think he had doubts as to the validity of SR - he said he did not think it would survive the test of time - the CBR is certainly different in different frames
Moreover, i do not think he would say, as to the two synced clocks which I described, where one is put in motion, that the one in motion would measure the non moving clock to be running slow (at least by the same factor) He Never said this - some authors do - others stop short of making this statement - we have never made this experiment - and until we make a freespace experiment that shows that a pion traveling at 0.99c relative to the Earth will measure Earth time to be slow, I think the question should remain unresolved - after all, relativity works fine whether or not all frames are perfectly equal. In short - I think the symmetry you demand does not comport with actual time dilation - it is consistent with apparent time dilation, and there is complete symmetry as to contaction - but as I have said - there is not complete symmetry when only one of two clock have been accelerated

 

[yogi]

Since I won't be able to follow pervects mathematical solution to the removing of a clock in orbit - I will propose the following - initially we have 3 clocks J, K, and L on the Earth - at the top of the tower - then we put two (J and K) in the same satellite and launch them into orbit - they should both run at the same speed and slower than the third clock (L) left atop the tower - then we decelerate (K) so that it comes to rest on the tower - it should now run at the same rate as the stay behind clock (L) since it has been returned to the original frame where it was synchronized

On the other hand, from the perspective of the J clock still in orbit - (K) has undergone an acceleration, and it should now run slower than J while J is in orbit. So we have returned to the original puzzle - The orbiting clock J runs slower than either K or L on top of the tower - but K should run slower than the orbiting clock J.