Talk:Test theories of special relativity

Sorry, This article Is Worthless

This is another poorly written un-understandable article that makes no sense and is confusing, vague, unclear, and essentially meaningless. Delete it or make it worthwile to read and entitled to be here.72.64.53.229 (talk) 15:37, 29 August 2008 (UTC)

Expansion

The article has been considerably increased by a description of Robertson's test theory, and the Standard Model extension. --D.H (talk) 15:56, 28 April 2011 (UTC)

Confusion

The article says that "b(v) is length contraction". However, b is defined (for SR) in such a way that it must be greater than 1 while the word "contraction" suggests that b is less than 1. Perhaps we should change this to "1/b(v) is length contraction"? After all, the distance in the moving frame should be obtained from the distance in the stationary frame by dividing (not multiplying) it by the shortened (contracted) measuring rod of the moving frame. JRSpriggs (talk) 03:58, 29 April 2011 (UTC)

They referred to the values in the modified Lorentz transformation, in which one must choose (p. 509) $1/a(v)=b(v)=1/{\sqrt {1-v^{2}}}$. Now, they also said, that "The coefficients 1/a(v) and b(v) are the time-dilatation and length contraction factors", which simply means the x- and t-transformations imply time- and length contraction. Maybe it should be formulated more elegantly by saying, that 1/a(v) and b(v) are the factors concerning time and length, so that the words "contraction" or "dilatation" are nor necessary anymore. --D.H (talk) 08:45, 29 April 2011 (UTC)

In that case, one could simply say "therefore a(v) is a factor related to time dilation and b(v) is a factor related to length contraction". If you are not going to be specific about the relationship, then there is no need for "1/" in combination with "a".

Notice that the time unit increases while the spatial unit decreases; that is, they change in opposite directions. This is due to using different kinds of side conditions, namely δx=0 (i.e. δX=vδT) for time while using δT=0 (not δt=0) for space. JRSpriggs (talk) 10:06, 29 April 2011 (UTC)

"Contraction by a factor of 2" means halving. "Dilation by a factor of 2" means doubling. On that basis I think it's correct to say "b(v) is length contraction", and "1/a(v) is time dilation".-- Dr Greg  talk  12:51, 29 April 2011 (UTC)

Calculating the speed of light in the moving frame

As in the article, we assume

$t=aT+ex\,$

$x=b(X-vT)\,$

$y=dY\,.$ (leaving z and Z out for simplicity)

Notice that symmetry gives us that a, b, d are even functions of v which must be one when v=0, so analyticity implies that to the second power of v

$a(v)\sim 1+\alpha v^{2}/c^{2}\,$

$b(v)\sim 1+\beta v^{2}/c^{2}\,$

$d(v)\sim 1+\delta v^{2}/c^{2}\,.$

Also e must be an odd function of v, so

$e(v)\sim \varepsilon v/c^{2}\,.$

Assuming an event on a light ray passing through the origin and since the angle θ should be measured in the laboratory (i.e. moving) frame, we get

$X^{2}+Y^{2}=c^{2}T^{2}\,$

$x=c't\cos \theta \,$

$y=c't\sin \theta \,.$

Now, calculating:

$T={\frac {t-ex}{a}}=t{\frac {1-ec'\cos \theta }{a}}\sim t{\frac {1-\varepsilon {v \over c^{2}}c'\cos \theta }{1+\alpha {v^{2} \over c^{2}}}}\sim t\left(1-\varepsilon {v \over c^{2}}c'\cos \theta -\alpha {v^{2} \over c^{2}}\right)$

$X={\frac {x}{b}}+vT\sim t{\frac {c'\cos \theta }{1+\beta {v^{2} \over c^{2}}}}+t\left(v-\varepsilon {v^{2} \over c^{2}}c'\cos \theta \right)\sim t\left(c'\cos \theta +v-(\beta +\varepsilon ){v^{2} \over c^{2}}c'\cos \theta \right)$

$Y={\frac {y}{d}}=t{\frac {c'\sin \theta }{d}}\sim t\left(c'\sin \theta -\delta {v^{2} \over c^{2}}c'\sin \theta \right)$

$T^{2}\sim t^{2}\left(1-2\varepsilon {v \over c^{2}}c'\cos \theta -2\alpha {v^{2} \over c^{2}}+\varepsilon ^{2}{v^{2} \over c^{4}}c'^{2}\cos ^{2}\theta \right)$

$X^{2}\sim t^{2}\left(c'^{2}\cos ^{2}\theta +2vc'\cos \theta -2(\beta +\varepsilon ){v^{2} \over c^{2}}c'^{2}\cos ^{2}\theta +v^{2}\right)$

$Y^{2}\sim t^{2}\left(c'^{2}\sin ^{2}\theta -2\delta {v^{2} \over c^{2}}c'^{2}\sin ^{2}\theta \right)\,.$

Substituting into the equation for the speed of light in the preferred frame and dividing by t² gives

$\left(c'^{2}\cos ^{2}\theta +2vc'\cos \theta -2(\beta +\varepsilon ){v^{2} \over c^{2}}c'^{2}\cos ^{2}\theta +v^{2}\right)+\left(c'^{2}\sin ^{2}\theta -2\delta {v^{2} \over c^{2}}c'^{2}\sin ^{2}\theta \right)$

$\sim c^{2}\left(1-2\varepsilon {v \over c^{2}}c'\cos \theta -2\alpha {v^{2} \over c^{2}}+\varepsilon ^{2}{v^{2} \over c^{4}}c'^{2}\cos ^{2}\theta \right)$

${c'^{2} \over c^{2}}+2{v \over c^{2}}c'\cos \theta -2(\beta +\varepsilon ){v^{2} \over c^{4}}c'^{2}\cos ^{2}\theta +{v^{2} \over c^{2}}-2\delta {v^{2} \over c^{4}}c'^{2}\sin ^{2}\theta$

$\sim 1-2\varepsilon {v \over c^{2}}c'\cos \theta -2\alpha {v^{2} \over c^{2}}+\varepsilon ^{2}{v^{2} \over c^{4}}c'^{2}\cos ^{2}\theta \,.$

By putting v=0, we see that to the zeroth power of v, $c'\sim c\,,$ so we can replace it in those terms which already have two factors of v so that to the second power of v

${c'^{2} \over c^{2}}+2{v \over c^{2}}c'\cos \theta -2(\beta +\varepsilon ){v^{2} \over c^{2}}\cos ^{2}\theta +{v^{2} \over c^{2}}-2\delta {v^{2} \over c^{2}}\sin ^{2}\theta$

$\sim 1-2\varepsilon {v \over c^{2}}c'\cos \theta -2\alpha {v^{2} \over c^{2}}+\varepsilon ^{2}{v^{2} \over c^{2}}\cos ^{2}\theta \,.$

Temporarily dropping terms with v², we get to the first power of v

${\left({c' \over c}\right)}^{2}+2(1+\varepsilon ){v \over c}\cos \theta \left({c' \over c}\right)-1\sim 0\,.$

Solving the binomial equation gives to the first power of v

${c' \over c}\sim -(1+\varepsilon ){v \over c}\cos \theta +{\sqrt {(1+\varepsilon )^{2}{v^{2} \over c^{2}}\cos ^{2}\theta +1}}\sim 1-(1+\varepsilon ){v \over c}\cos \theta \,.$

Substituting this into terms which already have one factor of v we get to the second power of v

${c'^{2} \over c^{2}}+2{v \over c}\cos \theta -2(1+\varepsilon ){v^{2} \over c^{2}}\cos ^{2}\theta -2(\beta +\varepsilon ){v^{2} \over c^{2}}\cos ^{2}\theta +{v^{2} \over c^{2}}-2\delta {v^{2} \over c^{2}}\sin ^{2}\theta$

$\sim 1-2\varepsilon {v \over c}\cos \theta +2\varepsilon (1+\varepsilon ){v^{2} \over c^{2}}\cos ^{2}\theta -2\alpha {v^{2} \over c^{2}}+\varepsilon ^{2}{v^{2} \over c^{2}}\cos ^{2}\theta \,.$

Rearranging terms

${c'^{2} \over c^{2}}\sim +1-2(1+\varepsilon ){v \over c}\cos \theta +2(1+\varepsilon ){v^{2} \over c^{2}}\cos ^{2}\theta +2(\beta +\varepsilon ){v^{2} \over c^{2}}\cos ^{2}\theta -{v^{2} \over c^{2}}+2\delta {v^{2} \over c^{2}}\sin ^{2}\theta$

$+2\varepsilon (1+\varepsilon ){v^{2} \over c^{2}}\cos ^{2}\theta -2\alpha {v^{2} \over c^{2}}+\varepsilon ^{2}{v^{2} \over c^{2}}\cos ^{2}\theta \,.$

${c'^{2} \over c^{2}}\sim +1-2(1+\varepsilon ){v \over c}\cos \theta +\left(+2+2\varepsilon +2\beta +2\varepsilon -1+2\varepsilon +2\varepsilon ^{2}-2\alpha +\varepsilon ^{2}\right){v^{2} \over c^{2}}$

$+\left(-2-2\varepsilon -2\beta -2\varepsilon +2\delta -2\varepsilon -2\varepsilon ^{2}-\varepsilon ^{2}\right){v^{2} \over c^{2}}\sin ^{2}\theta$

${c'^{2} \over c^{2}}\sim +1-2(1+\varepsilon ){v \over c}\cos \theta +\left(1+2\beta +6\varepsilon +3\varepsilon ^{2}-2\alpha \right){v^{2} \over c^{2}}$

$+\left(-2-2\beta +2\delta -6\varepsilon -3\varepsilon ^{2}\right){v^{2} \over c^{2}}\sin ^{2}\theta$

Applying the Taylor's series

${(1+w)}^{-{1 \over 2}}=1-{w \over 2}+{3w^{2} \over 8}-\ldots$

yields

${c \over c'}\sim +1+(1+\varepsilon ){v \over c}\cos \theta +\left(-{1 \over 2}-\beta -3\varepsilon -{3 \over 2}\varepsilon ^{2}+\alpha \right){v^{2} \over c^{2}}$

$+\left(1+\beta -\delta +3\varepsilon +{3 \over 2}\varepsilon ^{2}\right){v^{2} \over c^{2}}\sin ^{2}\theta +{3 \over 2}(1+\varepsilon )^{2}{v^{2} \over c^{2}}\cos ^{2}\theta$

${c \over c'}\sim +1+(1+\varepsilon ){v \over c}\cos \theta +\left(1-\beta +\alpha \right){v^{2} \over c^{2}}$

$+\left(-{1 \over 2}+\beta -\delta \right){v^{2} \over c^{2}}\sin ^{2}\theta$

If $\varepsilon =-1$ takes its SR value, then this simplifies to

${c \over c'}\sim +1+\left(1-\beta +\alpha \right){v^{2} \over c^{2}}$

$+\left(-{1 \over 2}+\beta -\delta \right){v^{2} \over c^{2}}\sin ^{2}\theta$

which is a rearrangement of the formula given in the article. JRSpriggs (talk) 02:06, 17 May 2011 (UTC)

Round trip experiments

Unfortunately, I haven't clearly written, that the formula

${\frac {c}{c'}}\sim 1+\left(\beta -\delta -{\frac {1}{2}}\right){\frac {v^{2}}{c^{2}}}\sin ^{2}\theta +(\alpha -\beta +1){\frac {v^{2}}{c^{2}}}$

refers to the (synchronization independent) round-trip speed, as measured by MM or KT (see the third paper by Sexl/Mansouri, or Giulini etc.). So there is no place for synchronization parameters in the section on experiments. --D.H (talk) 21:17, 16 May 2011 (UTC)

You caused me to lose my final edit on the section above. That was a lot of work down the drain. Please use the "new section" tab in the future when creating a new section instead of editing the end of the previous section. (Although I should have realized how to save it off-line, but I did not think of it.) I had converted the equation to one for c/c' and tried to determine whether changes were needed to incorporate varepsilon into the other terms of the equation. JRSpriggs (talk) 21:38, 16 May 2011 (UTC)

I actually have used the "New section" button, so I don't know why you have lost your data. --D.H (talk) 21:43, 16 May 2011 (UTC)

To D.H.: I am sorry that I expressed anger at you. I do not usually get so upset.

I grant that the term

$+(1+\varepsilon ){v \over c}\cos \theta$

will cancel out when you calculate the two-way speed of light because the cosine will change its sign but not its magnitude. However, the varepsilons which appear in the other terms will not cancel out. At this point, I do not know whether I made a mistake in my calculation or whether your claim that the two-way speed of light is independent of varepsilon is wrong. Can you prove the independence? JRSpriggs (talk) 02:50, 17 May 2011 (UTC)

I found my error. It was in solving the binomial equation. After making the corrections to the following formulas, the varepsilons in the other terms of the result disappeared. So if my calculation is now correct, then the two-way speed of light should indeed be independent of varepsilon. JRSpriggs (talk) 03:51, 17 May 2011 (UTC)

What assumptions are needed to justify the test theories?

The assumption that a preferred frame of reference has an isotropic speed of light is given explicitly, but I think that the test theories also assume implicitly that (in a preferred frame):

• the laws of physics are invariant under translation through time;
• the laws of physics are invariant under translation through space; and
• the laws of physics are invariant under spatial rotations.

These are the symmetries shared by Newtonian physics and SR. Only the symmetry under boosts is omitted. We also need to assume:

• the length of a rod is independent of its past and future, depending only on its structure, speed (not acceleration or higher derivatives), and the angle between its velocity and its length;
• all types of rods (or other distance measuring devices) are affected proportionately by speed;
• the rate at which a clock runs is independent of its past and future, depending only on its structure and speed;
• all types of clocks (or other duration measuring devices) are affected proportionately by speed;
• Newton's first law of motion (objects only accelerate in response to exogenous forces);
• the reference frame of an observer is constructed by him using these rods and clocks and Euclidean geometry and some (unspecified) procedure for synchronization of clocks which is independent of location, time, and orientation (relative to a preferred frame).

Should we not make these assumptions explicit in the article? JRSpriggs (talk) 16:52, 20 May 2011 (UTC)

I think the first thing the page should do is explain what a "test theory" is. Generally a theory is tested by using it alone to predict the outcome of an experiment. If the actual results match those predicted, the theory passes the test, if not it is falsified and other theories have no role to play in that process. The introduction says: "An experiment to test the theory of relativity cannot assume the theory is true .." but truth is not part of the scientific process, it is philosophical. I think some explanation of this is needed, for example saying that it is a way of classifying departures from Lorentz Invariance, something along the lines of PPN Formalism if that is a valid comparison. As it stands, it risks reading like pseudo-science written by some proposing an aether theory (e.g. Mansouri & Sexl). — Preceding unsigned comment added by GeorgeDishman (talk • contribs) 17:47, 11 October 2013 (UTC)