History of Topics in Special Relativity/Time dilation

History of time dilation
Time dilation in special relativity is given by


 * $$\Delta t'=\frac{\Delta t_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\approx\Delta t_{0}\left(1+\frac{v^{2}}{2c^{2}}\right)$$

Let a single clock A1 be at rest in inertial frame $$S$$ (indicating proper time $$\Delta t_{0}$$), and let two synchronized clocks B1 and C1 be at rest in inertial frame $$S'$$ (indicating Poincaré-Einstein synchronized coordinate time $$\Delta t'$$), with A1 traveling from B1 to C1 at speed $$v$$ (and A1 initially being synchronous to B1), then the formula tells us that when A1 reaches C1, the time indicated by A1 is lagging behind the time indicated by C1 by the Lorentz factor.

Since the laws of physics are the same in all inertial frames by the relativity principle, it follows that if one builds an experimental setup in which one has a single clock in $$S'$$ and two synchronized clocks in $$S$$, one gets a symmetrical result by simply exchanging the primes:


 * $$\Delta t=\frac{\Delta t'_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\approx\Delta t'_{0}\left(1+\frac{v^{2}}{2c^{2}}\right)$$

Let a single clock A2 be at rest in inertial frame $$S'$$ (indicating proper time $$\Delta t'_{0}$$), and let two synchronized clocks B2 and C2 be at rest in inertial frame $$S$$ (indicating Poincaré-Einstein synchronized coordinate time $$\Delta t$$), with A2 traveling from B2 to C2 at speed $$-v$$ (and A2 initially being synchronous to B2), then the formula tells us that when A2 reaches C2, the time indicated by A2 is lagging behind the time indicated by C2 by the Lorentz factor.

On the basis of electrodynamics, some sort of time dilation effect was described by Larmor (1897) regarding electron orbits and by Lorentz (1899) regarding oscillations. A more general description in terms of the “scale of time” was given by Larmor (1900, 1904), with Cohn (1904) specifically referring to moving clocks. The modern interpretation of time dilation as an observer dependent effect in terms of special relativity was given by Einstein (1905), while Minkowski (1907/08) provided the geometrical meaning in terms of proper time. The most common way to demonstrate time dilation is the so-called light clock, which was employed in the theory of the Michelson-Morley experiment by Michelson (1881-87) and Lorentz (1886) and was first used in relativity by Lewis and Tolman (1909).

Michelson (1881-87), Lorentz (1886)


Albert A. Michelson (1881) defined the travel time of the transverse beam in the w:first version of his famous experiment as $$2\tfrac{D}{V}$$ with D as arm length and V as speed of light. However, Michelson was informed by Alfred Potier that the motion of the apparatus also affects the transverse travel length, that is, his previous formula only applies to the case of rest in the aether. So Michelson published the correct travel length formula in 1882 as second order approximation:


 * $$2D\left(1+\tfrac{r^{2}}{2}\right)$$

where $$r=V/v$$ with v as relative speed between Earth and aether and V as speed of light. Finally, Michelson and Morley (1887) wrote that formula as follows:


 * $$2D\left(1+\tfrac{v^{2}}{2V^{2}}\right)$$

Also Hendrik Lorentz (1886) obtained the correct transverse travel time formula (which follows from Michelson's travel length formula by dividing with the speed of light) as second order approximation:


 * $$\frac{2D}{A}+D\frac{g^{2}}{A^{3}}$$

with D as arm length, A as speed of light, and g as relative speed between Earth and aether.

Larmor (1897-1904)
Joseph Larmor (1897), while discussing the Lorentz transformation between a system at rest in the aether and a moving one, concluded the existence of the following effect (where $$\epsilon=\left(1-v^{2}/c^{2}\right)^{-1}$$):


 * [..] the individual electrons describe corresponding parts of their orbits in times shorter for the latter system in the ratio $$\epsilon^{-1/2}$$ or $$\left(1-\tfrac{1}{2}v^{2}/c^{2}\right)$$, while those less advanced in the direction of v are also relatively very slightly further on in their orbits on account of the difference of time-reckoning.

While still based on electrodynamics, the somewhat broader term "scale of time" occurs in subsequent papers as follows:


 * 1900: [..] the electric and magnetic displacements at corresponding points of the fixed systems will be the values that the vectors [..] had at a time const. $$+vx/c^{2}$$ before the instant considered when the scale of time is enlarged in the ratio $$\epsilon^{1/2}$$.
 * 1904: [..] with the single addition of the FitzGerald-Lorentz shrinkage in the scale of space, and an equal one in the scale of time, which, being isotropic, is unrecognizable.

Larmor's contribution to time dilation was noticed by Wolfgang Pauli (1921):


 * There was first of all Larmor who, as early as 1900, set up the formulae now generally known as the Lorentz transformation, and who thus considered a change also in the time scale.

Lorentz (1899)
After Hendrik Lorentz (1892, 1895) introduced an "independent variable" called “local time” as a function of the location in the aether (in contrast to the “universal time”), he went on (1899, 1904) to define the exact Lorentz transformations between the aether and relatively moving systems, with the 1899 paper including the following statement:


 * [..] in S the time of vibration be $$k\varepsilon$$ times as great as in $$S_{0}$$. Now the number $$k\varepsilon$$ would be the same in all positions we can give to the apparatus; [..]

where $$k$$ is the Lorentz factor and $$\varepsilon$$ remains undetermined.

Poincaré (1900-1905)
Henri Poincaré in 1900 published the first operational interpretation to Lorentz's local time in terms of clock synchronization (to first order in v/c without time dilation), and in 1905 published the symmetric form of the Lorentz transformation (which makes the symmetry of time dilation obvious), though he didn't provide an operational description of time dilation in those early papers, and considered the "apparent duration" of clocks in motion only after 1905.

Cohn (1904)
Emil Cohn (1904) described the physical consequences of the transformations of Lorentz (1904) with respect to time as follows (where $$k$$ is the Lorentz factor):


 * The motions which are carried out by a material particle of the progressing system under the action of forces $$F$$ in space $$r$$ are different from the motions which the same particle in the case of rest is carrying out under the forces $$F_0$$ in space $$r_0$$, only by the fact that the process is slowed down in a constant ratio. Corresponding distances are traversed in times $$t$$ and $$t_0$$ connected by the equation $$t=kt_0$$


 * [..] $$t_0$$ are those time intervals indicated by an "initially correctly ticking" clock, after it was inserted into the system and accordingly has changed its rate.


 * [..] The Lorentzian interpretation requires from us, to distinguish between measured lengths and times $$x_{0}\dots t_{0}$$ and true ones $$x\dots t$$. But it doesn't give us the means to solve the task experimentally - even under presupposition of ideal measuring instruments. [..] Thus we have no other means at all, than to measure the distances with "false" co-moving measuring sticks, and to measure times with "falsely ticking" co-moving clocks.

Cohn's criticism of the Lorentzian distinction between "true" and "apparent" measurements was also mentioned by Moritz Schlick (1920) in a footnote in the third German edition of his book on Space and Time:


 * It deserves to be mentioned that already in the year 1904, E. Cohn published two treatises "On the electrodynamics of moving systems" in the proceedings of the Prussian academy of sciences, in which he sharply and clearly defended the claim of relativity of time- and space measurements against the Lorentzian interpretation (see the text above).

Einstein (1905)
Albert Einstein (1905) developed what is now called special relativity by recognizing that the Lorentz transformation is concerned with the nature of space and time and by abandoning the aether.

He showed that the relation between time t of a stationary system, and a clock at rest in a moving system k indicating time $$\tau$$, is given by:
 * $$\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}=t-\left(1-\sqrt{1-\left(\frac{v}{V}\right)^{2}}\right)t$$

So it is retarded by $$\left(1-\sqrt{1-(v/V)^{2}}\right)$$, or approximately $$\tfrac{1}{2}(v/V)^{2}$$, seconds per second. Then he derived the consequence, that if a clock is moved between synchronized clocks at rest in the stationary system from A to B, the moving clock will not by synchronous anymore at B, but retarded approximately by $$\tfrac{1}{2}tv^{2}/V^{2}$$. Einstein quickly concluded that the same holds even when the clock is moving on an arbitrary polygonal line between A and B, even if A and B coincide.

Minkowski (1907/8)
Hermann Minkowski (1907/8) clarified the role of time in the context of the spacetime formulation of special relativity as follows (with imaginary coordinate $$x_4 =it$$):


 * Transforming a space-time point $$P(x, y, z, t)$$ to rest is equivalent to introducing, by means of a Lorentz transformation, a new system of reference $$x', y', z', t'$$ in which the $$t'$$ axis has the direction OA', OA'  indicating the direction of the space-time line passing through P. The space $$t'=$$ const., which is to be laid through P, is the one which is perpendicular to the space-time line through P. To the increment dt of the time of P corresponds the increment:


 * $$d\tau=\sqrt{dt^{2}-dx^{2}-dy^{2}-dz^{2}}=dt\sqrt{1-\mathfrak{w}^{2}}=\frac{dx_{4}}{w_{4}}$$


 * of the newly introduced time parameter $$t'$$. The value of the integral


 * $$\int d\tau=\int\sqrt{-\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}\right)}$$,


 * when calculated upon the space-time line from a fixed initial point P° to the variable point P, (both being on the space-time line), shall be called the proper time of the position of matter we are concerned with at the space-time point P.

Lewis and Tolman (1909)
Gilbert N. Lewis and Richard C. Tolman (1909) illustrated time dilation as follows:


 * Let us consider two systems moving past one another with a constant relative velocity, provided with plane mirrors aa and bb parallel to one another and to the line of motion (Figure 1). An observer, A, on the first system sends a beam of light across to the opposite mirror, which is reflected back to the starting point. He measures the time taken by the light in transit.


 * A, assuming that his system is at rest (and the other in motion), considers that the light passes over the path opo, but he believes that if a similar experiment is conducted by an observer, B, in the moving system, the light must pass over the longer path mnm'  in order to return to the starting point; for the point m moves to the position m'  while the light is passing. He therefore predicts that the time required for the return of the reflected beam will be longer than in his own experiment. A, however, having established communication with B, learns that the time measured is the same as in his own experiment. [This is evidently required by the principle of relativity, for contrary to A's supposition the two systems are in fact entirely symmetrical. Any difference in the observations of A and B would be due to a difference in the absolute velocity of the two systems, and would thus offer a means of determining absolute velocity.]




 * The only explanation which A can offer for this surprising state of affairs is that the clock used by B for his measurement does not keep time with his own, but runs at a rate which is to the rate of his own clock as the lengths of the paths opo to mnm' .


 * B, however, is equally justified in considering his system at rest and A's in motion, and by identical reasoning has come to the conclusion that A's clock is not keeping time. Thus to each observer it seems that the other's clock is running too slowly.


 * This divergence of opinion evidently depends not so much on the fact that the two systems are in relative motion, but on the fact that each observer arbitrarily assumes that his own system is at rest. If, however, they both decide to call A's system at rest, then both will agree that in the two experiments the light passes over the paths opo and mnm' respectively, and that B's clock runs more slowly than A's. In general, whatever point may be arbitrarily chosen as a point of rest, it will be concluded that any clock in motion relative to this point runs too slowly.


 * Consider Figure 1 again, assuming system a at rest. We have shown that it is necessary to assume that B's clock runs more slowly than A's in the ratio of the lengths of the path opo to the path mnm' ; in other words, the second of B's clock is longer than the second of A's in the ratio mnm'  to opo. This ratio between the two paths will evidently depend on the relative velocity of the two systems v, and on the velocity of light c.


 * Obviously from the figure,


 * $$(op)^{2}=(ln)^{2}=(mn)^{2}-(ml)^{2}$$.


 * $$\frac{(op)^{2}}{(mn)^{2}}=1-\frac{(ml)^{2}}{(mn)^{2}}$$.


 * But the distance ml is to the distance mn as v is to c. Hence


 * $$\frac{mn}{op}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$.


 * Denoting the important ratio by the letter &beta;, we see that in general a second measured by a moving clock bears to a second measured by a stationary clock the ratio $$\frac{1}{\sqrt{1-\beta^{2}}}$$.