# Relativity on Earth

1. May 6, 2014

### analyst5

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?

2. May 6, 2014

### 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.

3. May 6, 2014

### 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.

4. May 6, 2014

### pervect

Staff Emeritus
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.

5. May 7, 2014

### 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?

6. May 7, 2014

### A.T.

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.

7. May 7, 2014

### Staff: Mentor

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.

8. May 7, 2014

### epovo

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

9. May 7, 2014

### analyst5

I'm sure this is completely wrong, not because I'm an expert in physics, but because the opposite answer has occured 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.

10. May 7, 2014

### 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?

11. May 7, 2014

### Staff: Mentor

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

12. May 7, 2014

### 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.

13. May 7, 2014

### analyst5

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.

14. May 7, 2014

### 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.

15. May 7, 2014

### Staff: Mentor

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.

16. May 7, 2014

### 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.

17. May 7, 2014

### analyst5

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)?

18. May 7, 2014

### Staff: Mentor

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

19. May 7, 2014

### 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.

Last edited: May 7, 2014
20. May 7, 2014

### D H

Staff Emeritus
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 Teph 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 Teph 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 Teph 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).

21. May 7, 2014

### pervect

Staff Emeritus
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.

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"

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 travelled 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.

22. May 11, 2014

### analyst5

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?

Last edited: May 11, 2014
23. May 11, 2014

### Staff: Mentor

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.

24. May 11, 2014

### WannabeNewton

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.

25. May 11, 2014

### epovo

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