Special Relativity - Reference Frames $S$ and $S'$

Question

Consider the standard arrangement in special relativity. Let $S'$ move in the +ve $x$-axis with a velocity $v$ with respect to $S$. This implies that $S$ moves with a velocity $-v$ with respect to $S'$.

Is this an assumption or a theorem of special relativity?

If theorem - How can it be derived?

@RogerJBlarlow below provided an answer with basically says $f(f(v))=v$ hence $f(v)= -v$. There are many other solutions to this so why do we only choose where $f(v)=-v$

This question has been puzzling me for a while now. Any help is much appreciated. Thanks

Answer 1

There is already a good answer by @Dale. Though if you want to do little math, you can do it this way. Write the Lorentz transformation $$x'=\gamma \ (x-vt)$$ $$t'=\gamma \left(t-\frac{vx}{c^2}\right)$$ Use these two find $(x,t)$ as function of $(x',t')$ which would lead to

---

$$x=\gamma \ (x'+vt')$$ $$t=\gamma \left( t'+\frac{vx'}{c^2}\right)$$ That's what you are asking for.

---

You can go even further through the basics remains the same. We know that co-ordinate transformations are just rotations in Hyperbolic space. Thus $S$ to $S'$ is a rotation through an angle, say $\beta$, then from $S'$ to $S$ is a rotation through an angle $-\beta$. If you recall $$\begin{pmatrix} x \\ ct \end{pmatrix}=\begin{pmatrix} \cosh\beta & \sinh\beta \\ \sinh\beta & \cosh\beta \end{pmatrix} \begin{pmatrix} x' \\ ct' \end{pmatrix}$$

---

Edit :

$$x'=\gamma \ (x-vt)\Rightarrow \frac{x'}{\gamma}+vt=x$$ Use this in $$t'=\gamma \left(t-\frac{vx}{c^2}\right)$$ $$\Rightarrow t'=\gamma \left( t-\frac{v}{c^2}\left(\frac{x'}{\gamma}+vt\right)\right) $$ Solve this for $t$ and you would find $$t=\gamma \left( t'+\frac{vx'}{c^2}\right)$$ In a similar manner, you can find the other equation for $x$ eliminating $t$.

Answer 2

It's a theorem which comes from the basic symmetry between S and S'.

Let $v$ be the relative velocity between O and O', as determined in S.

Transforming from S to S' gives some $v'=f(v)$

Transforming again from S' to S gives $v''=f(v')$, with the same $f$ because S from S' is the same as S' from S.

But we are now back where we started, so $v''=v$

Hence $f(f(v))=v$ for all $v$. This (given that $v$ has dimensions) means $f(v)=v$ and $f(v)=-v$ are the only possibilities. the former applies to the speed, the latter to the velocity.

Answer 3

This follows directly from the principle of relativity. The principle of relativity states that there is no way physically to distinguish between different reference frames. If the velocities were different then you could distinguish frames on the basis of their relative speeds. In principle there would in principle be a reference frame where the relative speed was minimal or maximal, and this frame would be a unique “privileged” frame.

Answer 4

Ever since 1910 the Russian physicist Vladimir Ignatowsky noticed that some assumptions about the nature of inertial frames (linearity, homogeneity, isotropy, reciprocity etc.) lead uniquely to either the Lorentz transformations of special relativity or to Galileo’s transformations of classical Newtonian mechanics.

The so-called reciprocity principle (i.e. the object of your question) is an axiom for the inertial reference frames: it can't be proved, but it's implicitly assumed in the definition of such reference systems.

On Journal of High Energy Physics volume 2012, Article number: 119 (2012) or also in the pre-print you can find an in-depth discussion of your question.

Answer 5

It is a postulate in my opinion.

Suppose there is an atomic clock in the Mars robot programmed to send signals every $\Delta t$. The radial velocity $v$ between earth and mars changes with time. The time between signals received at earth is:$$\Delta t_e = \Delta t + \frac{r+v\Delta t}{c} - \frac{r}{c} = \left(1 + \frac{v}{c}\right)\Delta t$$

Now suppose that another atomic clock at earth is sending signals at the same rate $\Delta t$. And the result of $\Delta t_m$ (that is informed to us) shows a persistent difference compared to $\Delta_e$.

I believe that the research, if any experimental errors were discarded, would target first the possibility that the average $c$ between the planets is not the same for both directions, due to some gravitational effects. Rather than assuming that $v$ is not symmetric.