Eisberg's 'Fundamentals of Modern Physics' derives the space contraction formula from a mirror experiment in which A reflects a light beam in a direction perpendicular to the motion of B, both observers measure the time and position of the beam's return to the path of motion. He needs this result to derive the LT; it uses the Pythagorean theorem to reach the gamma formula. Is it possible to derive the LT without making use of more than one space dimension?

In an experiment involving a train and two explosions that leave damage on both the train and the neighboring areas on the ground, he shows that if the explosions are simultaneous to a stationary observer, they cannot be to the train observer; the train observer will believe the explosion ahead of him happened sooner than the one behind him. But I cannot calculate gamma and the LT from this situation. It appears the extra spatial dimension of the mirror experiment is needed.

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    $\begingroup$ What do you want to derive the Lorentz transformations from here? Also, in a universe with only one spatial dimension, the Lorentz group is very different from the higher dimensions - it just has a single generator (i.e. no separate boosts and rotations) - so you should not expect to conclude anything about relativity in higher dimensions from it. $\endgroup$ – ACuriousMind Apr 7 '17 at 11:47
  • $\begingroup$ sure you can en.wikipedia.org/wiki/… $\endgroup$ – user126422 Apr 7 '17 at 12:10
  • $\begingroup$ I believe this might be closer to what you asked pa.msu.edu/courses/2000fall/PHY232/lectures/relativity/… $\endgroup$ – user126422 Apr 7 '17 at 16:59
  • $\begingroup$ or this: faculty.luther.edu/~macdonal/LorentzT.pdf $\endgroup$ – ZeroTheHero Apr 8 '17 at 0:53
  • $\begingroup$ See my answer here. In general, I think thinking about more than two spacetime dimensions in special relativity isn't really necessary or useful for someone learning it for the first time. $\endgroup$ – Abhimanyu Pallavi Sudhir Jan 26 '19 at 8:23

The mirror thought-experiment (due, I think, to Lewis and Tolman, c 1909) is, in my opinion, a brilliant way to confront beginners in Relativity with the astonishing consequences of Einstein's innocent-looking postulates. Aesthetically, though, I think it leaves something to be desired - for exactly the reason to which you draw attention: it calls on two spatial dimensions. Rindler (Introduction to Special Relativity, second edition) has an essentially one-dimensional derivation of the x and t Lorentz transforms, though he does carefully consider y and z, but separately.

Rindler argues carefully for transforms between x and x' and vice versa to be $$x'=g(x-vt),$$ $$x=g(x'+vt').$$ in which g is a constant, which is easily shown to equal the familiar gamma by considering a flash of light emitted from the origin, for which $x=±ct$ and $x'=±ct'$.

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Hermann Bondi encouraged the use of the k-calculus to derive the equations of special relativity in (1+1)-dimensions. That method uses radar measurement with light signals along the direction of relative motion (not transverse as in the textbook light-clock).

https://en.wikipedia.org/wiki/Bondi_k-calculus#Radar_measurements_and_velocity https://archive.org/details/RelativityCommonSense

(Physically, "k" is the Doppler factor. The method exploits the fact that k is the eigenvalue of the Lorentz transformation. The time-dilation factor is then obtained as $\gamma=\frac{k+k^{-1}}{2}$.)

Instead of the usual transverse light clock, one can use a "longitudinal light clock" to derive the equations of (1+1)-dimensional special relativity.

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You can derive the Lorentz Transform (LT) purely within 1-dimensional space using the geometry of a space-time diagram for two light transmissions along the x-direction only. The approach below is based on the book Special Relativity by AP French. But whereas French assumes (pg 78) at the outset a relation of form $x = ax' + bt'$, $x' = ax - bt$, for some $a$, $b$, the derivation below does not make any assumptions about the form of the transformation matrix - only that it is some linear transform, ie a general $2 \times 2$ matrix.

French justifies the linearity of the LT by arguing (pg 77) that a non-linear transform would not map a constant velocity to a constant velocity - which would contradict the inertial property of the frames. A non-linear map would be represented by curved $x'$ and $t'$ axes in Fig. 2 below so a straight line in $xt$-space would not in general be a straight line in $x't'$-space.

We use the idea of a 'skew' coordinate system $C'$ in 2D based on two arbitrary (non-parallel) vectors $\underline{\mathbf{a}}$ and $\underline{\mathbf{b}}$ as in Fig 1 below - note these are not necessarily unit vectors. The transform matrix mapping from $C$ coords (the original Cartesian $xy$-coord system) to the $C'$ coord system (ie $x'y'$-coords) is given by $[\begin{array}{cc} \underline{\mathbf{a}} & \underline{\mathbf{b}} \end{array}]^{-1}$ ie \begin{equation} \left( \begin{array}{c} x' \\ y' \end{array} \right) = \left[ \begin{array}{cc} \underline{\mathbf{a}} & \underline{\mathbf{b}} \end{array} \right]^{-1} \left( \begin{array}{c} x \\ y \end{array} \right) = A \left( \begin{array}{c} x \\ y \end{array} \right) \tag{1}\label{eq:skew-transform} \end{equation} and any $2 \times 2$ non-singular matrix $A$ can be represented this way by choosing $\underline{\mathbf{a}}$ and $\underline{\mathbf{b}}$ as the columns of $A^{-1}$.

In Fig 1, it is easy to visualize the $(x', y')$ coords of a point $P(x,y)$ by making the parallelogram construction shown.

enter image description here

In the space-time diagram in Fig 2 we have a frame $S'$ moving at $v$ wrt a frame $S$ in the standard setup. These are 1-dimensional frames with only an $x/x'$-axis. (Fig 3 depicts the situation with $v < 0$).

enter image description here

enter image description here

$A, B, C$ are at rest in $S'$ and at $t = 0$ light source $B$ sends light signals to $A$ and $C$. The blue world lines in Fig. 2 show how $S$ sees $A, B, C$ moving with time at speed $v$. Events $D, F$ are where the light signals (which have the green world lines) reach $A, C$ respectively.

For $S$, using the $xt$-coord system, event $D$ comes before event $F$, ie light reaches $A$ before it reaches $C$. We require an $x't'$-coord system for $S'$ for which the speed of light is $c$ in all directions, to satisfy Postulate 2. Thus whatever that system is it must consider events $D$ and $F$ to be simultaneous because the distance covered within $S'$ is the same in each direction. Because we are arguing the transform must be linear, it must correspond with some choice of $x't'$-axes in Fig 2, because it must correspond with a non-singular $2 \times 2$ matrix.

We consider constructing such a coord system using the method of Fig 1, using the coord axes $x'$ and $t'$ shown in Fig 2, ie. the basis vectors $\underline{\mathbf{a}}$, $\underline{\mathbf{b}}$ lie along the $x'$, $t'$ axes respectively. (We would shift these axes down so the origin coincided with that of the $xt$-axes). Note the $x'$, $t'$ axes are not physical axes, they are simply being used to represent a general linear transform $(x, t) \mapsto (x', t')$ on $\mathbb{R}^{2}$ as in Fig 1 - the advantage then is that this transform can be easily visualized via a parallelogram type construction as in Fig 1. Then in Fig 2 we can visualize the coordinates of any point event $(x, t)$ in $S$ when seen in the $(x', t')$ space of $S'$. For example, when two point events in S are joined by a straight line parallel to the $x'$ axis then they have the same $t'$ coordinate - when they are joined by a straight line parallel to the $t'$ axis then they have the same $x'$-coordinate.

The directions for the $x'$ and $t'$ axes shown in Fig 2 are suitable choices because :

  1. $A$ and $D$ must have the same $x'$ coord,
  2. $C$ and $F$ must have the same $x'$ coord, and
  3. light arrival events $D$ and $F$ must have the same $t'$ coord, ie be simultaneous in $S'$.

Thus the required transformation from $(x, t) \mapsto (x', t')$ is $L = [\begin{array}{cc} \underline{\mathbf{a}} & \underline{\mathbf{b}} \end{array}]^{-1}$ where $\underline{\mathbf{a}}$ has the gradient $m$ shown in Fig 2, and $\underline{\mathbf{b}}$ has the gradient $1/v$. (Fig 3 shows the case where $v < 0$). Other choices of gradients would not satisfy the conditions 1-3. This only defines the directions of $\underline{\mathbf{a}}$ and $\underline{\mathbf{b}}$ however, so we can write :

\begin{equation} L = [\begin{array}{cc} k_{1}\underline{\mathbf{a_{0}}} & k_{2}\underline{\mathbf{b_{0}}} \end{array}]^{-1} \tag{2}\label{eq:L-general} \end{equation}

where $$ \underline{\mathbf{a_{0}}} = \left( \begin{array}{c} 1 \\ m \end{array} \right), \underline{\mathbf{b_{0}}} = \left( \begin{array}{c} v \\ 1 \end{array} \right), $$

and $k_{1}$ and $k_{2}$ are arbitrary non-zero constants.

Any matrix $L$ of this form will satisfy the condition of making events $D$ and $F$ above simultaneous in $S'$.

The value of $m$ is easily found from the $(x, t)$ coords of $D$ and $F$. D is the intersection of lines $x = vt - l$ and $t = -x/c$ : \begin{eqnarray*} -ct & = & vt - l \\ \Rightarrow t & = & \frac{l}{c + v} \\ \Rightarrow x & = & \frac{-cl}{c + v} \\ \Rightarrow D & = & \frac{l}{c + v}(-c, 1). \end{eqnarray*} F is the intersection of lines $t = x/c$ and $x = vt + l$ : \begin{eqnarray*} ct & = & vt + l \\ \Rightarrow t & = & \frac{l}{c - v} \\ \Rightarrow x & = & \frac{cl}{c - v} \\ \Rightarrow F & = & \frac{l}{c - v}(c, 1). \end{eqnarray*} Then \begin{equation} m = m_{DF} = \frac{\Delta t}{\Delta x} = \frac{l\left( \frac{1}{c - v} - \frac{1}{c + v} \right)}{l\left( \frac{c}{c - v} + \frac{c}{c + v} \right)} = \frac{2v/(c^{2} - v^{2})}{2c^{2}/(c^{2} - v^{2})} = \frac{v}{c^{2}} \tag{3}\label{eq:calc-m} \end{equation}

We can determine $k_{1}$ and $k_{2}$ using the following symmetrical property of the relative velocity between two inertial frames $S$ and $S'$ : if $S$ sees $S'$ moving at constant velocity $v$ then $S'$ sees $S$ moving at constant velocity $-v$. (This is obvious under Classical Physics as we have absolute space and time, but under Special Relativity we have to assume it).

Firstly we can show $k_{1} = k_{2}$ :

(i) Consider space-time coords of $O'$. In $S'$, these are $(0, t')$ and the corresponding coords in $S$ are $(x, t)$. But since $S$ sees $O'$ moving at velocity $v$ we must always have $x/t = v$.

For now writing $$ L^{-1} = \left[ \begin{array}{cc} a & b \\ d & e \\ \end{array} \right] $$ we have \begin{eqnarray} x & = & ax' + bt' \nonumber \\ t & = & dx' + et' \tag{4}\label{eq:L-inverse} \end{eqnarray} and so for $O'$ which has $x' = 0$ for all time : \begin{eqnarray*} x & = & bt' \\ t & = & et' \end{eqnarray*} and then since also $x/t = v$ for all time, we have $b/e = v$.

(ii) Consider space-time coords of $O$. In $S$, these are $(0, t)$. And $O$ is moving at velocity $-v$ in $S'$, so $(x', t')$ for $O$ must always satisfy $x'/t' = -v$. Thus from (\ref{eq:L-inverse}), we have : \begin{eqnarray*} 0 & = & ax' + bt' \\ t & = & dx' + et' \end{eqnarray*}

the first of which implies $0 = a(-v) + b$, ie. $b/a = v$. Thus $v = b/e = b/a$ and so $a = e$. But from (\ref{eq:L-general}) the diagonal terms $a, e$ are just $k_{1}, k_{2}$, hence $k_{1} = k_{2} = k$, say.

(An alternative method of showing $k_{1} = k_{2}$ is to note the distance travelled by each light signal in $S'$ is represented by $\frac{1}{2}DF$, and its journey time in $S'$ is represented by $BE$, so that the speed of these signals in $S'$ is $c'$ given by : $$ (c')^{2} = \frac{ \left(\frac{1}{2}DF\right)^2 / (v^{2}/c^{4} + 1)k_{1}^{2} }{ BE^{2} / (v^{2} + 1)k_{2}^{2} }, $$ the scaling factor in the numerator being the length of the $k_{1}\underline{\mathbf{a_{0}}}$ basis vector, and the scaling factor in the denominator being the length of the $k_{2}\underline{\mathbf{b_{0}}}$ basis vector. Then using $DF^{2} = \Delta t^{2} + \Delta x^{2}$ ($\Delta t$ and $\Delta x$ taken from (\ref{eq:calc-m})), and $E = \frac{1}{2}(D + F)$, we can show $c' = c$ iff $k_{1} = k_{2}$, so that the latter follows from Postulate 2).

Next we show $k = \gamma$, where $\gamma$ is the Lorentz factor $1/\sqrt{(1 - v^{2}/c^{2})}$.

From (\ref{eq:L-general}) and (\ref{eq:calc-m}) we have : $$ L^{-1} = k \left[ \begin{array}{cc} 1 & v \\ v/c^{2} & 1 \\ \end{array} \right] $$ so $\det L^{-1} = k^{2}(1 - v^{2}/c^{2}) = k^{2} / \gamma^{2}$, and thus $$ L = \frac{\gamma^{2}}{k} \left[ \begin{array}{cc} 1 & -v \\ -v/c^{2} & 1 \\ \end{array} \right]. $$ This is the transform from $S$ to $S'$. It follows that since $S'$ observes $S$ moving at a relative velocity of $-v$ in the standard setup the transform $M$ from $S'$ to $S$ is this same matrix but with $v$ substituted by $-v$, ie. $$ M = \frac{\gamma^{2}}{k} \left[ \begin{array}{cc} 1 & v \\ v/c^{2} & 1 \\ \end{array} \right]. $$ But $M$ must equal $L^{-1}$, so now $$ \frac{\gamma^{2}}{k} \left[ \begin{array}{cc} 1 & v \\ v/c^{2} & 1 \\ \end{array} \right] = k\left[ \begin{array}{cc} 1 & v \\ v/c^{2} & 1 \\ \end{array} \right], $$ and hence $\gamma^{2} / k = k$, ie $\gamma^{2} = k^{2}$, so $k = \pm \gamma$. Choosing $k = -\gamma$ just gives essentially the same transform again but with the basis vectors multiplied by $-1$, so $$ L = \gamma \left[ \begin{array}{cc} 1 & -v \\ -v/c^{2} & 1 \\ \end{array} \right] $$ is the only possible linear transform that can satisfy Postulate 2 in one dimensional space.

When this is extended into three dimensional space by defining $y' = y$ and $z' = z$ we need to check the speed of light is preserved in all directions, and that speeds of less than $c$ are mapped to speeds of less than $c$ - and the Lorentz Speed Transform can be used to show this - this transform also shows constant speeds are not necessarily mapped to constant speeds. The Lorentz Velocity Transform shows constant velocity maps to constant velocity, but this also holds for any linear transform.

There are various intuitive arguments to justify setting $y' = y$ and $z' = z$, such as those involving rings or sticks passing one another, but in formal terms we can readily show the extension at least cannot be of the form $y' = \alpha y + \beta t$ and $z' = \eta z + \epsilon t$, unless $\alpha = 1, \beta = 0, \eta = 1, \epsilon = 0$ : https://physics.stackexchange.com/a/484294/111652.

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Rindler's and presumably Bondi's 1-dimensional methods work because they exploit spatial homogeneity, which Eisberg does not use. It seems constancy of c in 1 dimension is not enough to prove linearity of the LT; one needs either constancy in 2 dimensions, or homogeneity of space. I've never seen this cited before, that constancy of c in 2 dims is more powerful than in 1 dim.

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