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What gives you the idea that "A and B are not able t agree about the distance to that star"? The local speed of light is central to GR, it appears in the solution of the Einstein Field Equations (EFE). For example, it appears in the Schwarzschild solution: [math]ds^2=(1-r_s/r) c^2dt^2-dr^2/(1-r_s/r)[/math] Light follows null geodesics, so: [math]0=(1-r_s/r) c^2dt^2-dr^2/(1-r_s/r)[/math] From the above, you can get the {coordinate" (as opposed to "local") speed of light: [math]c_r=\frac{dr}{dt}=c(1-r_s/r)[/math] as I explained to you in a previous post. Looks like you want to "challenge" SR/GR. My advice to you is to start by studying.

In GR , there are no frames of reference covering the whole space, so, the notion of simultaneity can only be discussed locally. For a small enough region, the spacetime is locally Minkowski, so the GR notion of simultaneity reduces to the SR notion of simultaneity. For large® extents of space, nothing can be said about simultaneity. There is not much else that can be said on this subject.

A distant observer disagrees about the "coordinate" speed of light: [math]c_r=c(1-r_s/r)[/math] at points A and B. [math]r_s[/math] is the Schwarzschild radius of the Earth [math]c[/math] is the local speed of light (a universal constant) [math]r[/math] is the radial coordinate So, a distant observer measures: [math]c_A/c_B=(1-r_s/r_A)/(1-r_s/r_B)[/math] The "disagreement) is not 2 seconds, it is much less than that but this is irrelevant since: A and B agree on the "local" speed of light. Coordinate dependent entities are not meaningful in GR (and in physics, in general).

Yes, this is well known, as the "Sagnac effect".

The local speed of light is "c" even in rotating frames. There are preciously few papers on this subject but there are a few very good ones. So, esbo claim is false, he mixes the speed of light with closing speed. Closing speed is indeed [math]c \pm \omega r[/math], as evidenced by the class of experiments known as "Sagnac experiments".

Basic math says that the chain of equalities continues with [math]....=\gamma(t_a+v_2 x_a/c^2)-\gamma(t_a+v_1 x_a/c^2)=\gamma vx_a/c^2[/math] The arithmetic in posts 48 and 55 is pretty basic.

I assume that in the above "M" is the gravitational mass while "m" is the inertial mass. If so, your conjecture is many orders of magnitude off since it is well known (google "Eotvos experiment", "Einstein equivalence principle") that the two are equal to a very high order of precision.

******** Eighth Error: The correct equation, as already posted is: [math]t_{e2}-t_{e1}=\gamma(v_2) v_2x_a/c^2+t_a (\gamma(v_2)-1)=\gamma(v)vx_a/c^2+t_a (\gamma(v)-1) \approx \gamma(v) vx_a/c^2 [/math] As already explained, [math](\gamma(v)-1)t_a \approx \frac{v^2t_a}{2c^2} \approx 0[/math] so: [math]t_{e2}-t_{e1} \approx vx_a/c^2[/math] as already correctly explained in post 2

Actually, I am computing the correct extra term in the general formula explained in post 48. Nothing to do with any "time dilation". Let me tell you what I told Iggy, if you do not understand something, just ask, I will be more than happy to explain it to you.

The difference is that the website you linked is truly crackpot.

I don't understand why you have so much difficulty with the elementary exercises: [math]\gamma-1 =\frac{1}{\sqrt{1-(v/c)^2}}-1 \approx \frac{v^2}{2c^2}[/math] For [math]v=0.1m/s[/math] it follows that [math]\gamma-1 \approx 0.5* 10^{-19}[/math] so, for your chosen [math]t_a=1.5* 10^8[/math] [math](\gamma-1)t_a \approx 0.75* 10^{-11}[/math]

The above is a crackpot website written by a person that is in denial of mainstream physics.

*First Error: As explained [math](x_a,t_a)[/math] are the coordinates of the event in the Earth-Andromeda frame. **Second Error: There are only two velocities, not three, [math]v_1,v_2[/math]. [math]0,v[/math] are the values attributed to [math]v_1,v_2[/math] respectively. ***Third Error: There are only two frames, the Earth-Andromeda-Observer1 is one and Observer2 is the second one. The Lorentz transforms are quite trivial and self-explanatory. ****Fourth Error: This is outright false, the symbolic formulas work for any [math]t_a[/math]. These are the run of the mill Lorentz transforms, I do not understand why you have so much difficulty with the fact. *****Fifth Error: You obviously missed the fact that [math]t_a[/math] is multiplied by [math]\gamma(v)-1[/math] and [math]\gamma(v) \approx 1[/math]. It is all in the original post. ******Sixth Error: There is no reason for [math]t_a[/math] to be either zero or an ungodly large number. All is needed is to observe that [math]x_a[/math] is a very large (cosmological) distance such that it dwarfs [math]t_a[/math]. Combined with the fact that [math]t_a[/math] is multiplied by [math]\gamma-1 \approx 0[/math], you should get the idea.... *******Seventh Error: As explained, there is no reason for {math]t_a[/math] to be zero. It can be zero or it can be different from zero, the formalism I gave works either way.

You missed the trivial facts that [math]v_1=0[/math] , [math]v_2=v<<c[/math]. I don't know where you get the third variable for velocity but you are clearly wrong. Now, for time: you have the time for the Andromeda event ([math]t_a[/math]) and the two different coordinate times for the two Earth observers ([math]t_{e1},t_{e2}[/math]). This is a very simple exercise, I don't understand why you have so much difficulty with it. People who understand relativity know which speed, not velocity goes with [math]\gamma[/math]. Since you had so much difficulty, let me make it plain for you: [math]t_{e2}-t_{e1}=\gamma(v_2)(t_a+v_2 x_a/c^2)-\gamma(v_1)(t_a+v_1 x_a/c^2)=\gamma(v_2) v_2x_a/c^2+t_a (\gamma(v_2)-1)=\gamma(v) vx_a/c^2+t_a (\gamma(v)-1) \approx \gamma(v) vx_a/c^2[/math] [math]\gamma(v) \approx 1[/math] since [math]v<<c[/math] so: [math]t_{e2}-t_{e1} \approx vx_a/c^2[/math] as already correctly explained in post 2. Based on our prior interactions I thought you could do these simple calculations all by yourself. I would appreciate next time you challenge so hard and claim that you found mistakes that you ask first, I will be more than happy to explain to you.

Explanation is very simple: Say that two earthlings observe the event [math](x_a,t_a)[/math] on Andromeda. The earthlings are in relative motion with speed [math]v[/math] along what they call the [math]x-axis[/math] The observers have speeds [math]v_1=0[/math] respectively [math]v_2=v[/math] wrt Andromeda, so: [math]t_{e2}-t_{e1}=\gamma(t_a+v_2 x_a/c^2)-\gamma(t_a+v_1 x_a/c^2)=\gamma vx_a/c^2[/math]

Obviously, you have explained it incorrectly and I pointed out your mistakes. On a different note: the language of physics is math. Until you learn how to express yourself in a mathematical formalism, you aren't doing physics. English composition isn't physics. I think you are deeply confused, I post again, bolded this time, in the hope you will understand : Now, if one insists on having two observers, the math gets just a tad more complicated but the outcome is just the same. Say that two earthlings observe the event [math](x_a,t_a)[/math] on Andromeda. The earthlings are in relative motion with speed [math]v[/math] along what they call the [math]x-axis[/math] The observers have speeds [math]v_1=0[/math] respectively [math]v_2=v[/math] wrt Andromeda, so: [math]t_{e2}-t_{e1}=\gamma(t_a+v_2 x_a/c^2)-\gamma(t_a+v_1 x_a/c^2)=\gamma vx_a/c^2[/math] The math is very simple and the definitions are very precise, I do not understand why you have so much difficulty and you keep misrepresenting.

Thank you for (finally) admitting it. No one said that the explanation requires "extreme speeds" (BTW: 130km/s isn't extreme). What it was said, time and again, is that the paradox requires that [math]vx[/math] is of the order of [math]c^2[/math]. See my post 2, the one that generated all the flurry of answers. This is not how it is done, I have already shown you how to do this correctly: Now, if one insists on having two observers, the math gets just a tad more complicated but the outcome is just the same. Say that two earthlings observe the event [math](x_a,t_a)[/math] on Andromeda. The earthlings are in relative motion with speed [math]v[/math] along what they call the [math]x-axis[/math] The observers have speeds [math]v_1=0[/math] respectively [math]v_2=v[/math] wrt Andromeda, so: [math]t_{e2}-t_{e1}=\gamma(t_a+v_2 x_a/c^2)-\gamma(t_a+v_1 x_a/c^2)=\gamma vx_a/c^2[/math]

That is, of course, not quite correct. The mathematical formalism contains nothing about two observers as already shown in post 2. Only in the last section do the Wolfram people talk about two observers. This is clearly explained in this post .

I am used to negative feedback from people with a limited understanding of the issues, relativity is one of the disciplines that lends itself to such misinterpretations very often. As to your claim of "Misinterpretations of the Andromeda Paradox", I would like, once more , to direct your attention to the Wolfram webpage, generally they are very good at what they are doing. The fact that they use a somewhat unorthodox version doesn't mean that they are incorrect, it only means that the rest of the posters in this thread refuse to make an attempt at understanding the underlying physics. In order to help such people, I created post 27. I already did that in post 27, I explained both variants of the paradox for you and the others, I have directed you to reading it, twice.

The Op's question made absolutely no sense, so I don't see how you could give an answer. Would you mind trying to put the above in a mathematical formalism, after all , the language of physics is math. Ah, I see your "answer", you simply retaliated with feedback.

For people who are very fixated on a dogmatic interpretation of the paradox, I wrote post 27. In post 2 I presented the Wolfram variant, it seems that it riled a lot of people that refused to accept it as legit. <shrug> Good, can I recommend that you now read post 27? It gives you two different facets of the paradox. Wolfram used a somewhat unorthodox interpretation, perfectly correct nevertheless (you still don't understand it but this is ok). The whole thing is truly trivial, it is about the influence of large cosmological distance ([math]x[/math]) in the way distant observers (two or only one, according to Wolfram) evaluate the temporal separation [math]\gamma vx/c^2[/math].

I understand that dogma is difficult to overcome, I deal with this sort of attitude all the time. The people at Wolfram, would disagree with your statement. For people who are not that fixated in their thinking, I made post 27, perhaps you could take some time and read it.

Aren't these the exact words on the Wolfram website?

Why don't you make the effort to read the post just above yours? Then, you'll find out.

Quite a few posters have missed the fact that the above is a variant of the "Andromeda paradox", one with only one observer, NOT two. Nevertheless, the "paradox" works just the same and the above explanation is perfectly correct. Now, if one insists on having two observers, the math gets just a tad more complicated but the outcome is just the same. Say that two earthlings observe the event [math](x_a,t_a)[/math] on Andromeda. The earthlings are in relative motion with speed [math]v[/math] along what they call the [math]x-axis[/math] The observers have speeds [math]v_1=0[/math] respectively [math]v_2=v[/math] wrt Andromeda, so: [math]t_{e2}-t_{e1}=\gamma(t_a+v_2 x_a/c^2)-\gamma(t_a+v_1 x_a/c^2)=\gamma vx_a/c^2[/math] So, I solved a slightly different scenario, people familiar with relativity should have had no difficulty in realizing that the scenarios are equivalent. Instead, they posted repeatedly stuff that reflects their lack of understanding . Actually, this is false, the Wolfram page has only one observer. You can stop negging my posts now, I am simply explaining away your misconceptions.