Relativity on Earth: Understanding Simultaneity & Time Dilation

[Nugatory]

So the same can be said in the scenario where I'm sitting on the rotating Earth which is revolving around the sun, I just have to take the orbital velocity into account, right? In inertial frames, no matter what synchronization-parameter value that we choose between 0 and 1, moving clocks disagree about simultaneity with stationary ones?

Yes. However, the speeds involved are small enough that we never notice the deviations. So the theoretical flight time from London to Mumbai is a few nanoseconds different from the theoretical flight time in the other direction? The Hafele-Keating experiment didn't cause anyone to go off and start reprinting the airline flight schedules.

For daily life, we use (generally without realizing it) something like Einstein simultaneity (subtract the light travel time from the time the light reached our eyes) for things in our immediate vicinity and something equivalent to the TAI pervect described in #4 when we're working with widely separated events such as airline arrivals and departures. The discrepancies among them are real but so small that we neither notice nor care.

 

[Dale]

So is this convention the Einstein simultaneity for non-inertial frames that was previously mentioned, or is it something else?

It is something else. Usually the convention described is what I call "naive simultaneity" where simultaneity is determined by the momentarily co-moving inertial frame of the non-inertial observer. However, this runs into mathematical problems since it doesn't meet the requirement of being 1-to-1 everywhere, see: http://arxiv.org/abs/gr-qc/0104077

The closest thing to the Einstein simultaneity convention for non-inertial frames is the Dolby and Gull "radar time" convention in the linked paper, which I like but which is not a standard convention. Note that I don't say that the D&G convention is Einstein simultaneity since Einstein simultaneity is only for inertial observers, just that D&G is the closest.

Obviously, as you say, we are free to adopt a convention where all of the points I mentioned agree on simultaneity, but this also looks detabatable in my mind beacuse in the case of the rotating disc different points on the different distance from the radius have different velocities so I don't see how can they be synchronized at all?

Adopt the following synchronization convention: put a clock at the center, have it send out time-stamped light signals in all directions. At every radius, r, calculate the offset as $r/c$. Then at any point the synchronized time is the currently received time-stamp plus the offset.

Voila! All points on a rotating disk are synchronized. Note that this is NOT the Einstein synchronization convention. You clearly can synchronize points on a rotating disk, just not using Einstein's convention.

 

[analyst5]

Thanks for your last post, Dale.

Regarding the concept of 'naive simultaneity' that you previously mentioned, it seems that using it the distant clocks run faster than the closer ones during the 'sweeping phase' of the acceleration, but does each clock run at a constant rate during the acceleration of the moving twin. By that I mean if we have a clock that is located a great distance away from the moving twin that is accelerating, it will run faster, but will the rate of the clock remain constant and would not speed up during the acceleration phase of the twin?

 

[Dale]

Unfortunately, I don't know the answer to that. I have never worked it through in that level of detail.

 

[A.T.]

By that I mean if we have a clock that is located a great distance away from the moving twin that is accelerating, it will run faster, but will the rate of the clock remain constant and would not speed up during the acceleration phase of the twin?

I guess if you dose the acceleration right, then you can keep the rate of the distant clock (according to some simultaneity convention) constant during the acceleration.

But I also never worked it out.

 

[analyst5]

Ok, thanks to both of you. But it seems that this 'convention' is avoided in most cases. So regarding simultaneity from Earth's perspective, we can use the coordinate chart that we like with a reasonable definition, but no matter what simultaneity convention we use the relativistic effects and disagreements in simultaneity judgements for different observers under the same convention and for any observer at all will be extremely small for space near Earth since we are moving with very slow velocities compared to the speed of light? It doesn't matter if the system is moving inertially or non-inertially for the fact that at low speeds like on Earth the effects do exist, but aren't really big or noticeable?

 

[Nugatory]

no matter what simultaneity convention we use the relativistic effects and disagreements in simultaneity judgements for different observers under the same convention and for any observer at all will be extremely small for space near Earth since we are moving with very slow velocities compared to the speed of light? It doesn't matter if the system is moving inertially or non-inertially for the fact that at low speeds like on Earth the effects do exist, but aren't really big or noticeable?

Pretty much, yes.

There are a very few exceptions, such as the GPS system, where we do have to make relativistic corrections. The GPS system actually has its own formally defined simultaneity convention which it uses inside itself to coordnate the signals from the various GPS satellites. But whether you're driving 100 km/hr north or 100 km/hr south when you get your GPS position fix, the difference between your time and GPS time (after adding in your timezone) is completely negligible and we don't worry about it.

 

[Dale]

no matter what simultaneity convention we use the relativistic effects

When you are dealing with a non-inertial coordinate system then you need to determine the metric in that coordinate system. It will deviate from the Minkowski metric, but it will nonetheless contain all relativistic effects as well as any fictitious forces or other terms that are not due to relativity per se but simply due to the coordinate system.

 

[analyst5]

When you are dealing with a non-inertial coordinate system then you need to determine the metric in that coordinate system. It will deviate from the Minkowski metric, but it will nonetheless contain all relativistic effects as well as any fictitious forces or other terms that are not due to relativity per se but simply due to the coordinate system.

So what is the speed of light in non-inertial frames, or more specifically here on Earth? Is it also conventional (in a one way sense) like when choosing the synchronization parameter in inertial frames between 0 and 1 (which gives different values of the one way speed of light) , or does it differ in a different way?

 

[Dale]

So what is the speed of light in non-inertial frames, or more specifically here on Earth? Is it also conventional (in a one way sense) like when choosing the synchronization parameter in inertial frames between 0 and 1 (which gives different values of the one way speed of light) , or does it differ in a different way?

This gets to be a little tricky with the terminology.

Most of the time we use "reference frame" as a synonym for "coordinate system", but here is a place where the actual distinction makes a difference.

In GR, a reference frame is also known as a tetrad. It is a set of four orthonormal vector fields which represent local rods and clocks. Importantly, these rods and clocks are not synchronized or otherwise coordinated, so you can only make "local" measurements. The combination of orthonormality and local measurements ensures that the speed of light is still c in any reference frame, whether it is inertial or non-inertial.

On the other hand, a non-inertial coordinate system can use arbitrary synchronization conventions as well as arbitrary spatial conventions, so the speed of light can be arbitrary. You are correct that it is similar in principle to choosing the synchronization parameter, but even more general.

 

[analyst5]

This gets to be a little tricky with the terminology.

Most of the time we use "reference frame" as a synonym for "coordinate system", but here is a place where the actual distinction makes a difference.

In GR, a reference frame is also known as a tetrad. It is a set of four orthonormal vector fields which represent local rods and clocks. Importantly, these rods and clocks are not synchronized or otherwise coordinated, so you can only make "local" measurements. The combination of orthonormality and local measurements ensures that the speed of light is still c in any reference frame, whether it is inertial or non-inertial.

On the other hand, a non-inertial coordinate system can use arbitrary synchronization conventions as well as arbitrary spatial conventions, so the speed of light can be arbitrary. You are correct that it is similar in principle to choosing the synchronization parameter, but even more general.

Thanks for the good explanation. So if I'm understanding this well, the question 'what is the speed of light' on Earth (or what is the speed of light that comes to our eyes from everyday objects' does not have a definite and clear answer, and we may choose a different range of possibilites. Therefore different conventions may use a different light speed as some kind of 'basis' for defining the properties of their coordinate systems. Please correct me if I'm wrong.

 

[Dale]

Thanks for the good explanation. So if I'm understanding this well, the question 'what is the speed of light' on Earth (or what is the speed of light that comes to our eyes from everyday objects' does not have a definite and clear answer, and we may choose a different range of possibilites. Therefore different conventions may use a different light speed as some kind of 'basis' for defining the properties of their coordinate systems. Please correct me if I'm wrong.

Yes, except that it isn't a question of being on Earth or not, it is a question of using inertial frames/coordinates or not. I.e. even though it wouldn't be fixed to the surface of the earth, you can certainly use an inertial frame in your analysis, and then the question has a definite answer.

 

[xox]

But the surface of the Earth is not inertial, right? Surely there must be some convention for non-inertial observers that applies to Earth which is spinning and revolving around the Sun. And that also describes lengths and time dilations 'as viewed' from some point, or multiple points on Earth?

Correct, the Earth surface is rotating and revolving meaning that the labs used for experimentation are not inertial. Nevertheless, the effects of Earth's rotation/revolution ARE factored into the experiments used for testing SR. The test theories of SR (both Mansouri-Sexl and Standard Model Extension) take into account the exact motion of the Earth and its effects onto the canonical tests (Michelson-Morley, Kennedy-Thorndike and Ives-Stilwell). The bottom line is that the net effect of rotation is below any experimentally detectable level. This means, that for all intents and purposes, the Earth bound labs can be considered inertial. I believe that this was the gist of your question.

So what is the speed of light in non-inertial frames, or more specifically here on Earth? Is it also conventional (in a one way sense) like when choosing the synchronization parameter in inertial frames between 0 and 1 (which gives different values of the one way speed of light) , or does it differ in a different way?

So, it seems that I guessed right the gist of your questions. This is a VERY complicated question. In the test theories of SR, one way light speed is anisotropic and depends on a parameters, $$\alpha, \beta, \delta$$. More about it can be found in the famous papers by Mansouri and Sexl. Experimentally, though, the "anisotropy" is constricted to ever reducing limits, all tending towards zero. If you are interested, I can provide you with links to this subject.

 

[analyst5]

Correct, the Earth surface is rotating and revolving meaning that the labs used for experimentation are not inertial. Nevertheless, the effects of Earth's rotation/revolution ARE factored into the experiments used for testing SR. The test theories of SR (both Mansouri-Sexl and Standard Model Extension) take into account the exact motion of the Earth and its effects onto the canonical tests (Michelson-Morley, Kennedy-Thorndike and Ives-Stilwell). The bottom line is that the net effect of rotation is below any experimentally detectable level. This means, that for all intents and purposes, the Earth bound labs can be considered inertial. I believe that this was the gist of your question.
So, it seems that I guessed right the gist of your questions. This is a VERY complicated question. In the test theories of SR, one way light speed is anisotropic and depends on a parameters, $$\alpha, \beta, \delta$$. More about it can be found in the famous papers by Mansouri and Sexl. Experimentally, though, the "anisotropy" is constricted to ever reducing limits, all tending towards zero. If you are interested, I can provide you with links to this subject.

I think my level of understanding is below the information in the links you would provide me, but thanks for the help. It will make it easier if you could answer me what are the limits for the one-way speed of light in an inertial frame, when we change the synchonization parameter from 1/2 to some other value. I think it will help me to understand the potential differentiation between the one-way speed of light in a non-inertial frame. For instance, if we use 1/4 the return trip of the light is three times slower than the first leg. So in my calculation the speeds would be 450000 km/s and 150000 km/s in different directions. What are the lower and upper bounds when using this method?

And does a different simultaneity convention automatically apply a change in the one-way speed of light?

edit: what really makes this confusing is the fact that the speed of light may be so slow in some circumstances, that we perceive distant past even in the regions of space close to us.

 

[xox]

I think my level of understanding is below the information in the links you would provide me, but thanks for the help. It will make it easier if you could answer me what are the limits for the one-way speed of light in an inertial frame, when we change the synchonization parameter from 1/2 to some other value.

The link I provided shows that the most stringent limit is given by the modern re-enactment of the Michelson Morley experiment . It shows a deviation from the two-way light speed, "c" , of the order of 10^{-12}.

So in my calculation the speeds would be 450000 km/s and 150000 km/s in different directions. What are the lower and upper bounds when using this method?

This is WAY wrong. See correct results above.

 

[DrGreg]

I think my level of understanding is below the information in the links you would provide me, but thanks for the help. It will make it easier if you could answer me what are the limits for the one-way speed of light in an inertial frame, when we change the synchonization parameter from 1/2 to some other value.

The link I provided shows that the most stringent limit is given by the modern re-enactment of the Michelson Morley experiment . It shows a deviation from the two-way light speed, "c" , of the order of 10^{-12}.

So in my calculation the speeds would be 450000 km/s and 150000 km/s in different directions. What are the lower and upper bounds when using this method?

This is WAY wrong. See correct results above.

You are misunderstanding each other here. analyst5 is talking about 1-way speed and xox is talking about 2-way speed.

In a coordinate system where the 2-way speed is c, the 1-way speed could, with different sync conventions, take any value between ½c and ∞. If c₁ and c₂ are the 1-way speeds in opposite directions, they must be related by<br /> \frac{1}{c_1} + \frac{1}{c_2}= \frac{2}{c}<br />

 

[pervect]

You are misunderstanding each other here. analyst5 is talking about 1-way speed and xox is talking about 2-way speed.

In a coordinate system where the 2-way speed is c, the 1-way speed could, with different sync conventions, take any value between ½c and ∞. If c₁ and c₂ are the 1-way speeds in opposite directions, they must be related by<br /> \frac{1}{c_1} + \frac{1}{c_2}= \frac{2}{c}<br />

I'll go a bit further, and note that using the usual TAI or GPS time synchronization conventions (which are different than the Einstein convention!), on the surface of the Earth the one-way east-west coordinate speed of light is not equal to the one-way west-east coordinate speed.

I'm sure Doctor Greg already knows this, I'm trying to clarify things for readers like analyst5.

 

[xox]

You are misunderstanding each other here. analyst5 is talking about 1-way speed and xox is talking about 2-way speed.

Not at all, I was quite clear by pointing out to the test theories of SR and to the deviation of OWLS from TWLS , as being constrained by current experiments. I am clearly talking about OWLS. In fact, the test theories of SR (both M-S and SME) employ only two types of clock synchronization: Einstein and slow clock transport. Experimental tests based on either method of synchronization severely constrain the OWLS anisotropy , as I have already pointed out.

In a coordinate system where the 2-way speed is c, the 1-way speed could, with different sync conventions, take any value between ½c and ∞.

It could but, in actual experiments , it doesn't. As pointed out, the measured departure of OWLS from TWLS is of the order of 10^{-12}.

 

[DrGreg]

Not at all, I was quite clear by pointing out to the test theories of SR and to the deviation of OWLS from TWLS , as being constrained by current experiments. I am clearly talking about OWLS.



It could but, in actual experiments , it doesn't. As pointed out, the measured departure of OWLS from TWLS is of the order of 10^{-12}.

You've misunderstood what that Wikipedia article is saying. Using the notation of that article, the 1-way speed is determined by e(v). As the article says "The value of e(v) depends only on the choice of clock synchronization and cannot be determined by experiment."

The figure of 10^{-12} refers to the variation of the 2-way speed with direction (e.g. North-South v. East-West) which is what the Michelson-Morley experiment measured.

 

[xox]

You've misunderstood what that Wikipedia article is saying. Using the notation of that article, the 1-way speed is determined by e(v). As the article says "The value of e(v) depends only on the choice of clock synchronization and cannot be determined by experiment."

This means that all tests start by CHOOSING a synchronization method (either Einstein or slow clock transport), as I have already explained and PROCEED to constraining OWLS.

The figure of 10^{-12} refers to the variation of the 2-way speed with direction (e.g. North-South v. East-West) which is what the Michelson-Morley experiment measured.

I will have to respectfully disagree with you on this issue as well, all tests of light speed anisotropy refer to OWLS, not to TWLS.

Looking at the wiki article I can see what misled you, the sentence:

"Deviations from the two-way (round-trip) speed of light are given by:..."

\frac{c}{c&#039;}\sim1+\left(\beta-\delta-\frac{1}{2}\right)\frac{v^{2}}{c^{2}}\sin^{2}\theta+(\alpha-\beta+1)\frac{v^{2}}{c^{2}}should actually be corrected to read:

"Deviations of one way light speed, c', from the two-way (round-trip) speed of light ,c, are given by:..."

Now, the sentence: "The value of e(v) depends only on the choice of clock synchronization and cannot be determined by experiment." means exactly what it says, the experiments constraining OWLS do so by constraining \alpha, \beta , \delta as I explained earlier in the thread. The aforementioned experiments are incapable of constraining e. In fact, e is FIXED by the CHOICE of the synchronization method upfront, at the setup of the experiment.

 

[analyst5]

You are misunderstanding each other here. analyst5 is talking about 1-way speed and xox is talking about 2-way speed.

In a coordinate system where the 2-way speed is c, the 1-way speed could, with different sync conventions, take any value between ½c and ∞. If c₁ and c₂ are the 1-way speeds in opposite directions, they must be related by<br /> \frac{1}{c_1} + \frac{1}{c_2}= \frac{2}{c}<br />

So there's no way that the one way speed of light could go below 150000 km/s, no matter what synchonization parameter we use?

 

[xox]

So there's no way that the one way speed of light could go below 150000 km/s, no matter what synchonization parameter we use?

Actually, c_1,c_2 can take any value (if you use unphysical synchronization methods). Experiment tells us that c_1=c_2 if we use reasonable synchronization methods (either Einstein or slow clock transport). There is absolutely no reason to think that OWLS is anisotropic.

 

[Jorrie]

Actually, c_1,c_2 can take any value (if you use unphysical synchronization methods).

Only if you violate the isotropy of the two-way speed c, because as DrGreg said,

\frac{1}{c_1} + \frac{1}{c_2}= \frac{2}{c}

 

[xox]

Only if you violate the isotropy of the two-way speed c, because as DrGreg said,

\frac{1}{c_1} + \frac{1}{c_2}= \frac{2}{c}

Only if you violate the isotropy of ONE way light speed, not two-way light speed.
There is no experimental evidence supporting this concept, actually all experimental evidence supports the isotropy of OWLS.

 

[Nugatory]

Actually, c_1,c_2 can take any value (if you use unphysical synchronization methods).

Only if you violate the isotropy of the two-way speed c,

To the extent that two-way isotropy is experimentally confirmed (which is to say, pretty damned well), a synchronization method that leads to a violation of two-way isotropy is pretty much by definition unphysical, right? If so, you two are in violent agreement .

 

[xox]

To the extent that two-way isotropy is experimentally confirmed (which is to say, pretty damned well),

I am not aware of any experiments testing TWLS isotropy, all experiments I am aware of test OWLS isotropy, as I pointed out earlier in my response to DrGreg.

a synchronization method that leads to a violation of two-way isotropy is pretty much by definition unphysical, right?

Clock synchronization is tied to OWLS (actually, to the assumption that OWLS is isotropic, see Einstein synchronization), not to TWLS. It is true that OWLS isotropy results into TWLS isotropy. The reverse is not true, one can have TWLS isotropy without OWLS isotropy.

If so, you two are in violent agreement .

No, we are not in agreement. What Jorrie has posted doesn't even make sense (see above).

 

[xox]

I'll go a bit further, and note that using the usual TAI or GPS time synchronization conventions (which are different than the Einstein convention!), on the surface of the Earth the one-way east-west coordinate speed of light is not equal to the one-way west-east coordinate speed.

I'm sure Doctor Greg already knows this, I'm trying to clarify things for readers like analyst5.

With all respect I have to take exception to this statement. There are no global inertial frames of reference covering the whole circular trajectory of the GPS satellites. As such, there is no point in talking about the isotropy of coordinate light speed. Light speed is certainly isotropic LOCALLY, in a small interval along the trajectory. By contrast, over the WHOLE circle, the Sagnac effect MAKES IT LOOK AS IF the coordinate speeds in the opposing directions of circulation are different. This is a tremendous abuse of language, in reality we know is that what differs is the time taken by the em wavefronts to complete the circle. This effect, listed as the "Sagnac effect" in http://relativity.livingreviews.org/Articles/lrr-2003-1/fulltext.html is well known.

t_{\pm}=\frac{2 \pi R}{c \mp \omega R}

does not mean that the coordinate speed of the em wave has suddenly become anisotropic (c \mp \omega R). The mere notion of coordinate speed of light doesn't make sense in this case because of the absence of an inertial frame of reference covering the whole circle. Besides, one can argue that c \mp \omega R is technically the closing speed between the light front and the receiver, not the coordinate speed.

 

[Jorrie]

To the extent that two-way isotropy is experimentally confirmed (which is to say, pretty damned well), a synchronization method that leads to a violation of two-way isotropy is pretty much by definition unphysical, right? If so, you two are in violent agreement .

No, we are not in agreement. What Jorrie has posted doesn't even make sense (see above).

Then I think we are in non-violent disagreement...

Even ignoring the fact that the thread is about Earth's non-inertial frame and looking at a purely inertial frame, anisotropy of observed light propagation occurs when a non-standard synchrony is used. However, in any "sensible" non-standard synchrony, the observed two-way speed of light remains isotropic (because then synchronization of clocks does not play a role).

 

[DrGreg]

Experiment tells us that c_1=c_2 if we use reasonable synchronization methods (either Einstein or slow clock transport)

This seems to be the problem that you are failing to grasp. It's not experiment that tells us that, it's mathematical proof. If you use Einstein or slow clock transport it's mathematically guaranteed that c_1=c_2; no experiment required.

Experimental measurement of 1-way speed of light only makes sense if you are using some other sync method, and arguably the test is really whether the other method is equivalent to Einstein's or not.

 

[DrGreg]

So there's no way that the one way speed of light could go below 150000 km/s, no matter what synchonization parameter we use?

Provided the 2-way speed is c, and provided your sync method doesn't violate causality (i.e. doesn't allow signals to travel backwards in time), then yes.

It is possible to think up weird coordinate systems where those conditions might not be true, but not (if I understand correctly) in the context you were originally asking.