Relativity on Earth: Understanding Simultaneity & Time Dilation

[analyst5]

I posted about this on a muber of threads, but answers I red didn't really made me clear some confusion about this, so I apologize in asking about this but I'm just hoping of a convincing answer.

In talking relativity, most examples are given without any reference to our everyday lifes, since Earth is clearly a non-rotating frame which is rotating and undergoing circular motion around the sun. In inertial frames, the definition of simultaneity is standard and pretty clear. On the other hand, the definition of what events are simultaneous to anything on Earth is pretty vague, and the same applies to length contraction and time dilation here on Earth. So again, I apologize for bringing this up, but is there a clear example or convention that strictly applies to Earth which is rotating and circulating at the same time, or do we have to consider Earth as an isolated rotating system first and then as an isolating system that is revolving around the sun to define the relativistic effects?

Thanks in advance, analyst

 

[tom.stoer]

If one talks about proper time instead of coordinate time, then all the difficulties can be avoided. Think about the Haefele-Keating experiment, i.e. clocks orbiting around the Earth plus one "stationary" clock at the airport. After a full orbit you can compare proper times, taking into account rotation of the Earth and of the airplane as well as gravity. You can chose a non-rotating reference frame located at the center of the Earth in order to define space-time coordinates for the calculation, but of course the coordinate time will not show up when comparing proper times.

 

[BruceW]

I'm not totally sure what the question is. But I think most people would just choose an inertial frame of reference. For example, one which is rigidly attached to the sun. So then, if someone is born on the moon, and someone else is born on the earth, we can say (according to our reference frame attached to the sun), whether these events are simultaneous or not. You're right that if we choose a coordinate system that is non-inertial, we would have more work to do.

 

[pervect]

There are a LOT of different timescales in use, the topic isn't a simple one. There is a history of these different timescales evolving, as well. I'm not 100% sure I'll be up-to-date-to-the minute in this reply, but I'll try to give you an overview.

But the ultra-simple explanation, before I dive into the complexities, is that when you choose a coordinate system, that choice defines your notion of simultaneity. Hopefully you've figured out by now that this means that in the context of GR, simultaneity doesn't have direct physical significance, it's matter of convention. As long as you stick with your choice of coordinates, everything works out in the end - if you try and mix and match, you'll have more trouble, unless you carefully perform all necessary conversions.

The most commonly used definition of simultaneity on Earth would result from TAI time, international atomic time. This defines a coordinate time on the rotating Earth's surface, and a constant coordinate of TAI time would define simultaneity. IT's what you'd use in everyday life, but it isn't really suited to dealing with setting time coordinates any place other than the Earth's surface.

TAI time would also be what I think you were asking for, a coordinate time based on the Earth's surface suitable for "everyday life".

TAI time currently accounts for the effects of gravitational time dilation based on the Earth's gravitational field and height above the Earth's surface. It doesn't (AFIK) include for the gravity of the sun, or moon.

I'll briefly mention the time-system used for near-earth and solar system applications. For near-earth (think satellites), I believe they still use TCG, "Geocentric Coordinate Time", or the time derived from TCG, "TT" (Terrestial Time"). (Corrections are welcome).

One paper I've read says that there are two slight variants of TT, TT(BIPM) and TT(TAI).

On the solar system scale, we have TCB and TDB. TDB defined as a linear scaling of TCB adjusted in such a manner as to not diverge much from TT (which is tied to TAI - I *think*)

I hope this helps somewhat, at least to reveal how complex the topic is.

http://arxiv.org/abs/1208.3560 is of some interest, it describes a system of timing based on pulsars - it might help as an example how a time system like TAI that's well-suited for the purely terrestial applications might not be well suited for timekeeping for distant events.

 

[analyst5]

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?

 

[A.T.]

But the surface of the Earth is not inertial, right?

Yes, in relativity only free falling objects that undergo zero proper acceleration are inertial. The surface of the Earth undergoes 1g upwards proper acceleration. In terms of non-inertiality, the rotation of the Earth is a minor effect compared to that 1g acceleration. The surface of the Earth would be non-inertial even if it wasn't rotating.

 

[Dale]

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?

TAI is the most common convention, but you can propose and use other conventions as you like.

I am not sure what you mean by "as viewed", but if you don't feel that TAI meets that criterion then you are free to use your own.

 

[epovo]

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?

No, there is not. Simultaneity cannot be defined in non-inertial frames of reference.

 

[analyst5]

No, there is not. Simultaneity cannot be defined in non-inertial frames of reference.

I'm sure this is completely wrong, not because I'm an expert in physics, but because the opposite answer has occurred in multiple threads before by other members.


@Dale, Pervect
How are coordinates, distances and times defined in the TAI convention, and does it unify both the effects of rotation and circular non-inertial motion in regarding points on Earth as specific non-inertial frames.

 

[analyst5]

As I've red, the TAI convention accounts for GR effects, but if we solely focus on special relativity and non-inertial effects which can be handled by SR, what convention we might use for time on Earth? Marzke-Wheeler coordinates, which were mentioned in one of the previous threads?

 

[Dale]

does it unify both the effects of rotation and circular non-inertial motion in regarding points on Earth as specific non-inertial frames.

No. The Earth's rotation is not stable to the accuracy level of modern clocks, so the TAI is specifically designed to not fluctuate when the rotation of the Earth fluctuates. However, the clocks that define the standard are fixed to the surface of the earth, so they are time dilated according to relativistic effects including gravitational and movement based effects.

http://www.bipm.org/en/scientific/tai/tai.html

 

[epovo]

Let me make myself clear. If you have a number of clocks situated in several points (x,y,z) in a non-inertial frame, they will be ticking at different rates, so you cannot have a notion of simultaneity in a non-inertial frame.
A different story is having the TAI. The clocks are used to derive the TAI are fixed to the Earth surface, so they suffer from the same problem. BUT because we know the gravitational field and the rotation of the Earth we can make automatic corrections to the clocks so that they give a 'read-out' AS IF they were in an inertial frame.

 

[analyst5]

Let me make myself clear. If you have a number of clocks situated in several points (x,y,z) in a non-inertial frame, they will be ticking at different rates, so you cannot have a notion of simultaneity in a non-inertial frame.
A different story is having the TAI. The clocks are used to derive the TAI are fixed to the Earth surface, so they suffer from the same problem. BUT because we know the gravitational field and the rotation of the Earth we can make automatic corrections to the clocks so that they give a 'read-out' AS IF they were in an inertial frame.

The fact that there is no global simultaneity doesn't imply that each point on the surface has its local sense of simultaneity, right? So a non-inertial frame can have meaning only locally, and that's what I'm basically seeking. Except TAI, which looks pretty fine but uses GR elements, is there any convention strictly applicable just to SR, if we exclude gravitational time dilation.

In the rotating disk problem of SR, each point on the equator moves non-inertially and each point has a different sense of simultaneity, but if we add up the velocity that each point undergoes while circulating around the sun, what happens with the simultaneity judgements that we previously used? I'm clearly focusing here on the Earth in SR context, and seeking a coordinate chart that usefully combines the two different non-inertial movements of the Earth.

I hope someone can help me with this.

 

[epovo]

Two events are simultaneous in a reference frame if they both have the same t coordinate (I am not sure what you mean by local simultaneity, surely if two events happen on the same point and have the same t they are the same event!).
Let's take that "point on the surface" that you are referring to and make it the origin of our reference frame. Now you need to define your coordinates, i.e. a way to label every event with t,x,y,z. The rotating disk is the best thought experiment, because it's conceptually easy (although it's extremely complicated and challenged the best minds for a long time). And does not involve GR (no gravity). It's easy to fix a frame to the disk (let's say we pick the center of the disk as origin) and get x, y and z for every event. But how do you assign t to an event? If you think about it, you will soon realize you just can't. For example, in an inertial frame you can use the 'slow-moving clocks' concept to define simultaneity, but here that does not work.

 

[Dale]

Let me make myself clear. If you have a number of clocks situated in several points (x,y,z) in a non-inertial frame, they will be ticking at different rates, so you cannot have a notion of simultaneity in a non-inertial frame.

Sure you can. You just define your simultaneity convention in any way convenient. As a result of your simultaneity convention some of the clocks will be ticking slow and some will be ticking fast. This is time dilation.

 

[epovo]

My apologies. Of course, defining a simultaneity convention is the same as assigning a t coordinate to each event in your reference frame. You cannot have Einstein synchronization, but you can have a different sync convention. You can use Markze-Wheeler's construction of a reference frame, which has some desirable properties for rotating frames.

 

[analyst5]

Sure you can. You just define your simultaneity convention in any way convenient. As a result of your simultaneity convention some of the clocks will be ticking slow and some will be ticking fast. This is time dilation.

That's what I was referring to, as you said once Dale, you may use a non-standard convention. But what conventions are available, since Earth undergoes not one, but two different versions of non-inertial motion (rotation and revolution)?

 

[Dale]

But what conventions are available

TAI, UTC, etc. If you don't like them then you are free to define your own.

 

[WannabeNewton]

analyst the statements about simultaneity not being defined in non-inertial frames are entirely wrong so ignore them. Even Einstein simultaneity can be defined locally in non-inertial frames unless, roughly speaking, there is rotation of the frames in which case local Einstein simultaneity fails to hold (more precisely we say the synchronization gauge corresponds to a non-integrable connection form) but other conventions can be adopted.

As for your situation, you seem to be making things more complicated for yourself by considering highly complex realistic systems (physics is usually not done that way!). Consider instead a rigidly rotating disk that is also circularly orbiting a central point. Say the orbit radius is very large compared with the disk radius. Then the orbital motion is irrelevant to the problem of clock synchronization on the disk itself. This brings us down to just synchronization of clocks on a rigidly rotating disk. Einstein synchronization won't work for this obviously, because of the rotation. The easiest synchronization convention then is simply obtained by setting the times of all clocks on the disk to that of the central clock. On Earth this is what DaleSpam already mentioned: TAI, except now it also takes into account gravitational effects.

 

[D H]

I'll briefly mention the time-system used for near-earth and solar system applications. For near-earth (think satellites), I believe they still use TCG, "Geocentric Coordinate Time", or the time derived from TCG, "TT" (Terrestial Time"). (Corrections are welcome).

One paper I've read says that there are two slight variants of TT, TT(BIPM) and TT(TAI).

On the solar system scale, we have TCB and TDB. TDB defined as a linear scaling of TCB adjusted in such a manner as to not diverge much from TT (which is tied to TAI - I *think*)

I hope this helps somewhat, at least to reveal how complex the topic is.

Corrections:
The BIPM has farmed out the trickier aspects of what "time" is to the International Earth Rotation and References Services (IERS, and yes, the acronym doesn't make sense any more.) (web site www.iers.org). The IERS keeps track of time as a joint effort between the Paris Observatory and the US Naval Observatory (USNO), with a bit of "help" from JPL and the Russian Academy of Sciences. The Brits? They're not part of the equation. GMT no longer exists as far as the IERS is concerned.

TAI is perhaps the simplest. It is conceptually what an ideal atomic clock at sea level will read. It's not so simple in practice. Some atomic clocks are better timekeepers than others, but none are ideal. To get around these issues, the USNO collects data from a number of atomic clocks around the world, massage those data on a monthly basis, and then say what TAI was a month ago. Various researchers are investigating using entanglement to improve the precision and accuracy of atomic clocks.

TT is another supposedly simple time scale. The official IERS/BIPM definition is that TT is a fixed offset (32.184 seconds) from TAI. Except it's not. While the USNO and IERS don't make after-the-fact changes to TAI, they do make after-the-fact changes to TT.

Next is TDB, sometimes known as WTF. The French had their uniquely French way of representing TDB. JPL and the RAS preferred to disagree. For a long time, JPL's T_eph differed markedly in scale and offset with the official definition of TDB (which nobody used except the French). TDB has since been redefined to more or less coincide with the JPL DE405 T_eph as a time scale that on average ticks at the same rate as TAI but also reflects how time changes relativistically due to the varying depth of the Earth in the Sun's gravity field. In the meanwhile, JPL has released a number of DE models that improve upon DE405. JPL's T_eph varies from release to release. Being a standards organization, the BIPM/IERS definition of TDB hasn't changed a bit since that massive change where JPL and the RAS won (and the French lost).

 

[pervect]

I'm sure this is completely wrong, not because I'm an expert in physics, but because the opposite answer has occurred in multiple threads before by other members.


@Dale, Pervect
How are coordinates, distances and times defined in the TAI convention, and does it unify both the effects of rotation and circular non-inertial motion in regarding points on Earth as specific non-inertial frames.

The slightly oversimplified version would go like this. You first observe that all clocks on the "Geoid", which you can think of as sea level, run at the same rate. Then you need a mechanism to synchronize these clocks.

The study of methods to synchronize clocks is called "Time and Frequency transfer". NIST has a number of techniques listed, I'm not sure which one they are currently using:

See http://tf.nist.gov/time/gps.htm and http://www.tf.nist.gov/general/museum/847history.htm

None of the particular methods is particularly hard to understand, but there is a lot of detail in reading about all of them.

TAI time accounts for rotation of the Earth - it doesn't currently account for effects due to solar or lunar tides at the current level of precision.

TAI time doesn't have a "solar system view", it's restricted (by design and definition) to the surface of the Earth (and nearby points). So it doesn't (and can't) take into account effects of the Earth's orbit, in the TAI view, the Earth isn't moving, it's at the center of the coordinate system.

One of the biggest differences between TAI time and TCB time is the difference between having a view of time where the coordinate system origin is at the Earth (TAI) vs a view of time with the coordinate system centered at the Solar System barycenter (TCB). I'll give a reference here, because the discrepancy is interesting, though it may be distracting from the main point of understanding in depth what TAI time is and how it's implemented.

http://articles.adsabs.harvard.edu/abs/1967AJ...72.1324C

Back to the main point, explaining TAI time in depth.

You can think of TAI time (and also , distances) as being realized by the GPS system, and as being defined by the particular "mapping" or "metric" that the GPS system uses.

See http://arxiv.org/abs/gr-qc/9508043 "Precis of General Relativity"

A method for making sure that the relativity effects are specified correctly (according to Einstein’s General Relativity) can be described rather briefly. It agrees with Ashby’s approach but omits all discussion of how, historically or logically, this viewpoint was developed. It also omits all the detailed calculations. It is merely a statement of principles.

One first banishes the idea of an “observer”. This idea aided Einstein in building special relativity but it is confusing and ambiguous in general relativity. Instead one divides the theoretical landscape into two categories.

One category is the mathematical/conceptual model of whatever is happening that merits our attention. The other category is measuring instruments and the data tables they provide.

To paraphrase some of the detailed discussion that Misner gives, the mathematical conceptual model of "what is happening" is derived from a particular metric, or "map", of space-time, near the Earth's surface.

Why do I call a metric a "map", you ask?

The idea is the same, a "map" is just a one-one correspondence between points on a representation of a territory, and the territory itself.

In the case of the metric, the representation is more abstract than a piece of paper, but it serves much the same purpose, and the one-one correspondence between points on the "map" and points on the "territory" remains.

Using this metric, one can compute measured quantities such as the paths of light beams, _proper_ time readings (where the times are read on the ame clock), or anything else that one cares to measure.

Because our ability to measure time is more precise than our ability to measure distance, distances will typically be measured as proper times - for instance, if we want a distance of a meter, it's defined as "The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.".

I have skipped over corrections for things like delays due to the presence of matter (i.e. the absence of a vacuum), assuming that the measurements are processed to remove things like the delays due to the atmosphere _before_ one compares the measurement to the "map" that is created by the metric.

So to try and sum it all up - relativity predicts that measurements will be compatible with a certain metric, or map. The metric is realized mathematically, but it's not much different than using a globe or any other sort of representation of reality.

The metric allows one to calculate anything of interest. Specifically, the metric, plus some fixed reference objects (the GPS satellites) allows one to find "where" on the map one is, much as one might take bearings on several landmarks to determine where one was on a traditionally 2-d map on a sheet of paper. So the metric, plus some known reference objects, operationally defines the coordinates. We don't need anything to define coordinates, other than the metric and some reference objects that everyone agrees on.

Very often the fine level of detail of what reference objects one uses to locate oneself on a map is omitted, it is just trusted that the person trying to locate themselves on the map "does a good job".

Thus I don't really want to imply that it's absolutely necessary that the GPS satellites are the reference objects, but they're a typical implementation choice.

 

[analyst5]

analyst the statements about simultaneity not being defined in non-inertial frames are entirely wrong so ignore them. Even Einstein simultaneity can be defined locally in non-inertial frames unless, roughly speaking, there is rotation of the frames in which case local Einstein simultaneity fails to hold (more precisely we say the synchronization gauge corresponds to a non-integrable connection form) but other conventions can be adopted.

As for your situation, you seem to be making things more complicated for yourself by considering highly complex realistic systems (physics is usually not done that way!). Consider instead a rigidly rotating disk that is also circularly orbiting a central point. Say the orbit radius is very large compared with the disk radius. Then the orbital motion is irrelevant to the problem of clock synchronization on the disk itself. This brings us down to just synchronization of clocks on a rigidly rotating disk. Einstein synchronization won't work for this obviously, because of the rotation. The easiest synchronization convention then is simply obtained by setting the times of all clocks on the disk to that of the central clock. On Earth this is what DaleSpam already mentioned: TAI, except now it also takes into account gravitational effects.

So in the case you mentioned, all points on the disk will agree on simultaneity, despite their different rotational velocities and the fact they are also undergoing circular motion? Are we free to 'eliminate' the orbital motion when talking about defining a simultaneity convention for rotating bodies that also circulate?
Thanks for the answer, I'm sorry if I'm using a complicating scenario but honestly I'm tired in a way while learning about relativity because I don't have any knowledge about how to use it on our planet and what do its effects look like from our perspective.

edit: and how can the orbital motion be irrelevant in this case, if it's adding to the velocity of each rotating point on the disc?

 

[Dale]

Are we free to 'eliminate' the orbital motion when talking about defining a simultaneity convention for rotating bodies that also circulate?

You are free to do whatever you want when defining your convention as long as your resulting coordinates are smooth and 1-to-1. You don't even need to define a timelike coordinate at all, you can use null and spacelike coordinates exclusively if you like.

There is a standard convention for inertial reference frames, but once you step outside of that you have a lot of freedom, and it is simply a matter of definition.

Of course, the Christoffel symbols will be non-zero, but that is unavoidable for non-inertial coordinates anyway.

 

[WannabeNewton]

So in the case you mentioned, all points on the disk will agree on simultaneity, despite their different rotational velocities and the fact they are also undergoing circular motion?

Yes because I am setting the hands of all clocks on the disk in synchrony with that of the central clock. This synchronization procedure is perfectly valid for these clocks. The rotation of the disk is an issue when trying to get the clocks to be in Einstein synchrony-it is here that such matters fail to have the clocks on the disk agreeing in simultaneity (namely the Einstein synchronization fails to be transitive in the rotating frame of the disk).

edit: and how can the orbital motion be irrelevant in this case, if it's adding to the velocity of each rotating point on the disc?

If the disk radius is much smaller than the orbital radius (as is also the case with the Earth) then to a good approximation the points on the disk all have the same orbital velocity and so renders the orbital motion irrelevant to the problem of synchronizing only the clocks on the disk.

 

[epovo]

Yes because I am setting the hands of all clocks on the disk in synchrony with that of the central clock.

If it helps with visualization, the way I imagine this is having a lot of clocks at rest with respect to the center's clock hovering over the disk. A disk rider will assign a t coordinate to an event by glancing up at a nearby clock, ignoring the watch he's wearing

 

[analyst5]

Yes because I am setting the hands of all clocks on the disk in synchrony with that of the central clock. This synchronization procedure is perfectly valid for these clocks. The rotation of the disk is an issue when trying to get the clocks to be in Einstein synchrony-it is here that such matters fail to have the clocks on the disk agreeing in simultaneity (namely the Einstein synchronization fails to be transitive in the rotating frame of the disk).



If the disk radius is much smaller than the orbital radius (as is also the case with the Earth) then to a good approximation the points on the disk all have the same orbital velocity and so renders the orbital motion irrelevant to the problem of synchronizing only the clocks on the disk.

But if the disk wasn't even rotating, if it was just undergoing orbital motion, different points on the disc would still disagree about simultaneity despite having the same orbital velocity? Is this relevant with combining rotating and orbiting velocity? Thanks for the answer, but this still remains one of the most confusing aspects in SR for me.

 

[Dale]

But if the disk wasn't even rotating, if it was just undergoing orbital motion, different points on the disc would still disagree about simultaneity despite having the same orbital velocity?

IT DEPENDS ON THE CONVENTION YOU CHOOSE!

How many different times and different ways do you need to be told? Once you are dealing with non-inertial frames there is no standard convention so there is no standard answer. You can make the answer be whatever you want by picking a different convention.

 

[analyst5]

IT DEPENDS ON THE CONVENTION YOU CHOOSE!

How many different times and different ways do you need to be told? Once you are dealing with non-inertial frames there is no standard convention so there is no standard answer. You can make the answer be whatever you want by picking a different convention.

Here's the conceptual problem for me. When I think of non-inertial frames, the first thing that comes to my mind is the twin paradox and the circular motion that the moving twin undergoes while changing frames. In the classic case scenario during circular motion the twin quickly sweeps over the other twin's worldtube, and that's his definition of simultaneity during the non-inertial phase of motion. Now, as you say Dale, this is not the only possible convention.

If we generalize that convention to any kind of circular motion, be it orbital or rotational, the points on different positions in space that are undergoing non-inertial motion will disagree on simultaneity. So is this convention the Einstein simultaneity for non-inertial frames that was previously mentioned, or is it something else? When using this type of simultaneity definition, the points on the rotating disc disagree on simultaneity, and also different points on the object undergoing circular motion disagree on simultaneity. And if we combine these two motions we get an even bigger disagreement. That's why this confuses me. 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?

 

[Nugatory]

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?

They can't be, at least not in any universally accepted standard way. That's why there is no standard convention for simultaneity in this non-inertial frame.

I'm sitting on the rotating disk and looking at the universe around me. Three space coordinates and one time coordinate suffice to name any event out there, and I can assign these coordinates pretty much any way that I please, as long as I don't try to give two events separated by a time-like interval the same time coordinate, or two events separated by a space-like interval the same position coordinates (and a few other requirements). Two events are simultaneous if and only if I assign them the same time coordinate - but I can assign the time coordinates any way that I want (or not at all), so I can make it come out any way I want.

 

[analyst5]

They can't be, at least not in any universally accepted standard way. That's why there is no standard convention for simultaneity in this non-inertial frame.

I'm sitting on the rotating disk and looking at the universe around me. Three space coordinates and one time coordinate suffice to name any event out there, and I can assign these coordinates pretty much any way that I please, as long as I don't try to give two events separated by a time-like interval the same time coordinate, or two events separated by a space-like interval the same position coordinates (and a few other requirements). Two events are simultaneous if and only if I assign them the same time coordinate - but I can assign the time coordinates any way that I want (or not at all), so I can make it come out any way I want.

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?

 

[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.