Nature of Time Dilation and Length Contraction

You are only thinking of a single satellite, perhaps an equatorial orbit. You have 24 clocks in 6 orbital planes, all of which must beat at the same rate. Simply defining a circle as viewed from a different location does not keep the relative velocity of the satellite the same at apoapsis and periapsis of a POLAR orbiting satellite, for example. Multiple moving clocks can only be synchronized if all their orbits are circular, and all are orbiting a common coordinate point at the same velocity, the center of the Earth in the GPS example. This IS the ECI frame of reference. THEN, the ECEF frame of reference MUST have the same center-of-earth coordinates as the ECI frame. All position calculations must be done in the ECEF frame, not the ECI frame. The ECI frame is used only for clock synchronization due to velocity of the satellites relative to a non-rotating, earth-centered reference frame. All other corrections, such as gravitational time dilation, Sagnac effect, etc., are done in the ECEF frame of reference.

 

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Tom2,

If we assume that spacetime is a physical entity I can understand how it might be possible for matter to curve it. After all, the curvature of spacetime caused by mass is a local interaction. A person can clearly see the cause/effect relationship between matter and the curvature of spacetime, and can imagine a mechanism that is responsible for it.

However, I do not understand how an observer's relative motion can effect spacetime in the observer's entire frame of reference (which, by the way, includes the entire universe). I don't see the cause/effect relationship, nor do I see a mechanism that can make this possible. Do you? Do you find the curvature of spacetime in GR as real as the length contraction of spacetime in SR? Of course, both are mathematically possible, but are the both physically possible?

 

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This is incorrect (according to SR). Multiple moving clocks can be synchronized as long as their positions are known and the speed of light is constant. If clock A receives a signal at sa=(cta,xa,ya,za) in some inertial frame which was broadcast from clock B at sb=(ctb,xb,yb,zb) (where ta and tb are measured by the different clocks) in that same inertial frame then the clocks are synchronized in that inertial frame iff |sa-sb|=0. Circular orbits are easier to track and easier to keep synchronized and accurate, but not essential. The ECI frame is also easier to use, but not essential.

Here is a simplified but concrete example:

Let's say that we have two satellites in circular orbits in different orbital planes. They might have worldline equations in the ECI like

sa = (ct, sin(wt), cos(wt), 0)
sb = (ct, cos(wt), 0, sin(wt))

The north pole in the ECI is at p=(0, 0, 0, r) where r is the radius of the earth. So, in a pole-centered frame the equations of motion would be

sa' = sa-p
sb' = sb-p

The positions are still known and c is still constant in the pole-centered frame so the clocks can still be synchronized.

If we wanted to use a boosted frame then there would be some Lorentz transform L from the ECI to the boosted frame. Then the equations of motion would be:

sa' = L.sa
sb' = L.sb

Again, the positions are still known and c is still constant in the boosted frame so the clocks can be synchronized.

-Dale

 

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No. Based on your previous post I thought it was pretty clear that you wanted to start an irrelevant comment contest. Frankly, I am not sure who came out ahead there.

Wrong. v3= (v2-v1)/(1-v2 v1) would be the SR relative velocity between the clocks. However, why would we want to calculate that? Both sets of clocks are (approximately) in rotating frames so c is not constant in their frames. Therefore we want to synchronize each set of clocks with hypothetical colocated inertial clocks. For that, the relevant velocity is relative to the inertial frame, not the other clocks. It is not that difficult to grasp.

ROFL!

-Dale

 

The question remains. Do you agree that (other than topological elevation changes and crust density anomalies) the earth's geoid (oblate speroid) results in SR and GR affects cancelling such that over the surface clocks tick at a common rate? Yes or No?

Technically correct but using the velocity added formula (as you very well know) does not alter the result because these velocities are sub-subluminal.

I'll make you happy and recalculate using velocity addition. Then perhaps you will address the issue instead of trying to dodge it. :bugeye:

Using the VAF V3 = 3,410.7001953125m/s instead of just 3,410.7m/s. You still want to ignore the -5.8us/day SR figure vs the actual GPS figure of -7.2us/day using a local preferred common rest frame instead of SR?

Correction: I had posted from memory and the -5.8us/day according to SR is incorrect it is -5.583779us/day. That versus -5.583780us/day using your veloicty addition formula value for V3. However hardly changes the fact that the correct value is -7.2us/day based on using absolute veloicty to a local common preferred rest referance instead of SR's relative velocity. (It is absolute in the sense that it has no reciprocity, not that the value is absolute).

BTW are my calculations correct?

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Yes. I have read that the GPS system was designed that way and in my personal experience the GPS system works, so why would I disagree.

I know, I was just incredibly amused how you were boasting about "the magnitude of correctly posted calculations" when you didn't even have the formula right.

I didn't dodge it, but since your reading is apparently as poor as your math I will be glad to re-post it

-Dale

 

I would have no idea. But that was the inferance of your post.

I think you are desperate. NOBODY in their right mind would use velocity addition at such subluminal values. You are being silly.

Funny how you can post "I didn't dodge it" but then dodge it in the same post.

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************ Correction using Dale's velocity addition value *********

Using the VAF V3 = 3,410.7001953125m/s instead of just 3,410.7m/s. You still want to ignore the -5.8us/day SR figure vs the actual GPS figure of -7.2us/day using a local preferred common rest frame instead of SR?

Correction: I had posted from memory and the -5.8us/day according to SR is incorrect it is -5.583779us/day. That versus -5.583780us/day using your velocity addition formula value for V3.

An increase by 1.000000179/1. Picky, picky. Can you imagine had I pulled this on one of you?

However that hardly changes the fact that the correct value is -7.2us/day based on using absolute velocity to a local common preferred rest referance instead of SR's relative velocity. (It is absolute in the sense that it has no reciprocity, not that the value is absolute).

BTW are my calculations correct?
****************************************************

Wiggle, wiggle. We await.

 

I don't know how you infered my opinion on GR v. SR effects from a comment on socks.

OK, here goes take 3:

What exactly do you think was the design goal of all of the timing correction factors used in GPS?

-Dale

 

Not on your best day. When I introduced the GPS issues you were all over the map (wrong BTW) and not once corrected anything I had to say.

You bet he was. FYI he and I were in general agreement with a couple of symmantical exceptions. I clearly knew 6 times what you knew and he knew 6 times what I knew but wht I knew was generally corect and what you tried to claim was outright BS.

Outrightly false. Liar.

You have claimed GPS orbit to be inertial and you have also claimed they were accelerating frames. You have argued your "Subset to GR" view before and it doesn't hold water. Not that it isn't a subset of GR but that it must therefore be valid.

GR restores the preferred view and eliminates reciprocity. SR is about relative velocity between clocks. Surface clocks at the equator or poles have the same tick rate because SR and GR cancel. The velocity compensation made is for the absolute orbit velocity using the ECI FOR.

(absolute in this context means no reciproicty, not in value).

Good since my jpoint hasn't changed and has (and still does properly describe GPS. Yours hasn't).

I do. Get a life James R

 

You choose to ignore the content of my post. I will ignore yours.

 

Stil at at Mac!?? I'm waiting for your Book to come out...

 

ROFL! Reading your posts is so much fun. I respond each time to your point and you try to pretend each time that I didn't, and then you claim that I am the one ignoring the content of your posts. You really have exchanged sense for non-sense. This is truly amusing.

-Dale

 

Ditto.

Ditto.

Never has been any issue on my part. You were not teaching you were (as you are here) trying to find anything you could say to give the appearance of correcting my views. Doing so you have posted statements which are completely contridictary.

And no I am not wasting my time seaching through 4 years of crap to prove my assertion since you seem to feel you can make any false statement against me without doing the same.

Any such requirements is a two way street.

Rhetoric.

A and B are parts of a theory. A says it is Red, B says it is green; since we have pictures of it being red, it must therefore also be green.

You cannot advocate validity of a theory which is based on faulty physics.

Are we clear?

Preferred frame in the sense that GR has no reciprocity. Back with some referances.

 

Huh?

In GR, when a scenario contains one participant who has accelerated, and one participant who has not accelerated, the one who has NOT suffered acceleration is given the preferred title of prime, or, stationary.

As much as I have a personal phobia against another rehash of Dreaded Twin Paradox, Einstein himself claimed that the DTP, created by SR, was solved by GR with its mechanism for selecting, on the basis of who accelerated and who did not, the preferred frame of stationary observer, or participant.

Of course, it may depend on what definition of "preferred" is preferred.

 

This is only in theory i.e. if positions would be determined directly by comparison of the satellite time signal with a ground clock. In practice however positions are determined by differences of the time signal for different satellites. This means that there is actually no accumulation of positional errors due to the different rate of clocks in orbit. Taking the usually quoted a clock rate difference due to Relativity of 38 microseconds/day, this translates into a positional error of merely 1 cm, which is even less than the presently claimed accuracy of the GPS system (see my page Global Positioning System (GPS) and Relativity for more).

Anyway, I am not so sure if one should necessarily trust information that comes from the military (as they are still running the GPS system). For obvious reasons, they surely won't release all the information they have.

Thomas

 

Still collecting mis-statements from the experts.

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Considering that this is at least the 3rd or 4th time I've posted this question and your responses have been negative innuendo, I'll post it again and see just how long you choose to ignore thie issue.

Funny how you can post "I didn't dodge it" but then dodge it in the same post.

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************ Correction using Dale's velocity addition value *********

Using the VAF V3 = 3,410.7001953125m/s instead of just 3,410.7m/s. You still want to ignore the -5.8us/day SR figure vs the actual GPS figure of -7.2us/day using a local preferred common rest frame instead of SR?

Correction: I had posted from memory and the -5.8us/day according to SR is incorrect it is -5.583779us/day. That versus -5.583780us/day using your velocity addition formula value for V3.

An increase by 1.000000179/1. Picky, picky. Can you imagine had I pulled this on one of you?

However that hardly changes the fact that the correct value is -7.2us/day based on using absolute velocity to a local common preferred rest referance instead of SR's relative velocity. (It is absolute in the sense that it has no reciprocity, not that the value is absolute).

BTW are my calculations correct?
****************************************************

 

I took a quick look at your page, tsmid. What you are effectly calculating is a change in the speed of light due to frequency shift, then calculating the difference in travel times over 20000 meters based on the difference in travel speeds. Think this through again. Light travels almost 300 meters in one microsecond. In 38 microseconds, light will travel 11,400 meters. If there is a descrepency of 38 microseconds between the clocks, the calculated distance will be off by 11,400 meters. IIRC, the GPS clocks are synchronized to within about 42 nanoseconds per day of UTC time (other than leap seconds, which corrections for are included in the signal). This precise time is broadcast by the satellites for precision users on Earth, including the Arecibo radio telescope in Puerto Rico. Or at least, Arecibo used to use GPS signals, I believe they have recenty gone to using pulsars for precise timing purposes.