How does the Kennedy-Thorndike experiment test for time-dilation?

The Kennedy–Thorndike experiment, first conducted in 1932 by Roy J. Kennedy and Edward M. Thorndike, is a modified form of the Michelson–Morley experimental procedure, testing special relativity.[1] The modification is to make one arm of the classical Michelson–Morley (MM) apparatus shorter than the other one. While the Michelson–Morley experiment showed that the speed of light is independent of the orientation of the apparatus, the Kennedy–Thorndike experiment showed that it is also independent of the velocity of the apparatus in different inertial frames. It also served as a test to indirectly verify time dilation – while the negative result of the Michelson–Morley experiment can be explained by length contraction alone, the negative result of the Kennedy–Thorndike experiment requires time dilation in addition to length contraction to explain why no phase shifts will be detected while the Earth moves around the Sun.

The passage says that the original MM experiment had equal length arms (with respect to the rest frame) and the modification that comes with the KT experiment is that one of the arms is shortened. The passage claims that the KT experiment is able to test for time dilation as well as length contraction whereas the MM experiment only tests for length contraction. I understand how MM tests for length contraction (from the point of view of a relatively moving inertial observer), but how does KT test for time dilation? I would be interested in understanding this in more detail.

The way the KT experiment tests for time-dilation is similar to how the light-clock thought experiment can reveal time-dilation. They're not exactly the same, but I would argue the principles are very similar.

The reasoning here requires us to first accept Lorentz contraction. The MM experiment alone does not prove Lorentz contraction, because there are a number of possible theories that can account for the results of MM (dragged aether, ballistic/emission light theory), but a combination of late 19th century and early 20th century experiments do manage to narrow down on the fact that objects are Lorentz contracted when moving relative to some fixed inertial reference frame. With this in mind, let us take Lorentz contraction for granted.

To simplify things, I will also make the assumption that the lengths perpendicular to velocity are unaffected. This was apparently a hidden assumption of Kennedy and Thorndike when they claimed they could derive the full Lorentz transformation. See this passage in Wikipedia. (In fact, it is worth pointing out that Voigt transformations are an example that fits the MM and KT experiments but the time-dilation factor is different than the one we derive in this post.)

Part 1

First, let us consider a simplified warm-up thought experiment. There will be issues concerning conventionality and relativity of simultaneity, so in order to avoid them and sweep them under the rug, I will fix a reference frame $$F$$ as the so-called stationary inertial reference frame, and I will analyze everything with respect to $$F$$. Moreover, I will assume the speed of light is constant and isotropic in $$F$$.

Let us consider a variant of the simplified MM/KT setup where we have an interferometer with two arms of lengths $$L_{1}$$ and $$L_{2}$$ that are $$90^{\circ}$$ apart, and the apparatus is moving in the direction of arm $$\#1$$ at speed $$v$$ according to $$F$$. For simplicity, let us start by analyzing the pure kinematics of the situation. We can analyze this with light-rays or point-particles; we'll choose to consider light-rays.

Let $$C_{I}$$ (interferometer clock) be a clock placed right at the beam-splitter that moves with the interferometer, and let $$C_{S}$$ (stationary clock) be a clock that is stationary with respect to $$F$$.

Suppose a light-ray is shot. When it meets the beams-splitter it splits into two rays going through the two arms. Assuming Lorentz contraction occurs with respect to $$F$$, the arm lengths change: $$L_{1}\rightarrow L_{1}/\gamma(v)$$ and $$L_{2}\rightarrow L_{2}$$. By doing some basic kinematics math, we find that the light-ray in arm $$\#1$$ takes $$t_{1}(v) = 2L_{1}\gamma(v)/c$$ time to go from the beam-splitter and back, and light-ray going in arm $$\#2$$ takes $$t_{2}(v) = 2L_{2}\gamma(v)/c$$ according to $$C_{S}$$. Thus $$C_{S}$$ measures a time difference of $$\Delta t_{S}(v) = t_{1}(v) - t_{2}(v) = \frac{2\gamma(v)}{c}(L_{1}-L_{2}).$$ Now what does clock $$C_{I}$$ measure? The results of the KT experiment indicate (this needs an explanation which is provided below) that the time difference measured by $$C_{I}$$ is \begin{align}\tag{*} \Delta t_{I}(v) = \frac{2}{c}(L_{1}-L_{2}). \end{align} If this is true, then it follows that from the point of view of frame $$F$$ clock $$C_{I}$$ runs slower, and $$\Delta t_{S}(v) = \gamma(v)\Delta t_{I}(v).$$ This is precisely the statement of time-dilation.

Part 2

Ok, but Kennedy and Thorndike didn't have access to atomic clocks in 1932, so how could you do this without atomic clocks? They may not have had access to atomic clocks, but just like Michelson and Morelay, they could take advantage of light interference to tease out timing differences.

We will relate light interference and clock timings to each other in the following way. Let $$C_{I}$$ be a clock placed right at the beam-splitter that moves with the interferometer, let $$C_{I}'$$ be a clock placed right at the light source that moves with the interferometer, and let $$C_{S}$$ be a clock that is stationary with respect to $$F$$. Suppose the setup is moving at speed $$v$$ in direction of arm $$\#1$$ wrt $$F$$.

Let $$C_{I}'$$ measure the time period $$\Delta T_{I}(v)$$ of each wave-cycle of the light leaving the light source. Let $$C_{I}$$ measure the difference $$\Delta t_{I}(v)$$ between the two times it takes for light to make a round-trip in each of the arms. We also let $$\Delta T_{S}(v)$$ and $$\Delta t_{S}(v)$$ be the respective times as measured by clock $$C_{S}$$.

We make the assumption that clock $$C_{I}'$$ and the light source, if they are slowed or sped up at all wrt $$F$$, are slowed or sped up by the same factor wrt $$F$$. Thus, \begin{align}\tag{1} \Delta T_{I}(v) = \Delta T_{I}(0) \end{align} for all $$v$$. Further, we make the assumption that clocks $$C_{I}$$ and $$C_{I}'$$ are slowed or sped up by the same factor wrt $$F$$, which implies that both are slower or faster than $$C_{S}$$ by the same factor. Consequently, we have \begin{align}\tag{2} \frac{\Delta T_{I}(v)}{\Delta T_{S}(v)} = \frac{\Delta t_{I}(v)}{\Delta t_{S}(v)} \end{align} for all $$v$$. Also, we point out that at $$v=0$$, clocks $$C_{I}$$, $$C_{I}'$$, and $$C_{S}$$ all tick at the same rate, so \begin{align}\tag{3} \Delta T_{S}(0) = \Delta T_{I}(0). \end{align} By the same reasoning as in Part 1, $$\Delta t_{S}(v) = \frac{2\gamma(v)}{c}(L_{1}-L_{2})$$ for all $$v$$. In particular, \begin{align}\tag{4} \Delta t_{S}(v) = \gamma(v)\Delta t_{S}(0) \end{align} for all $$v$$.

As the light splits at the beam-splitter, makes its round-trips, and returns to the beam-splitter, the resulting phase difference is $$\Delta\phi(v) = \frac{\Delta t_{S}(v)}{T_{S}(v)}.$$ The result of the KT experiment is that $$\Delta\phi(v)$$ does not depend on $$v$$. In particular, we have $$\Delta\phi(v) = \Delta\phi(0)$$, so then \begin{align}\tag{5} \frac{\Delta t_{S}(v)}{\Delta T_{S}(v)} = \frac{\Delta t_{S}(0)}{\Delta T_{S}(0)} \end{align}

Now we will put all the equations $$(1)$$-$$(5)$$ together. By $$(5)$$ and $$(4)$$, $$\gamma(v) = \frac{\Delta T_{S}(v)}{\Delta T_{S}(0)}.$$ By $$(3)$$, this is $$\gamma(v) = \frac{\Delta T_{S}(v)}{\Delta T_{I}(0)}.$$ By $$(2)$$, this is $$\gamma(v) = \frac{\Delta T_{I}(v)\cdot \Delta t_{S}(v)/\Delta t_{I}(v)}{\Delta T_{I}(0)} = \frac{\Delta T_{I}(v)}{\Delta T_{I}(0)}\cdot\frac{\Delta t_{S}(v)}{\Delta t_{I}(v)}.$$ By $$(1)$$, this is $$\gamma(v) = \frac{\Delta t_{S}(v)}{\Delta t_{I}(v)},$$ and we arrive at the equation $$\Delta t_{S}(v) = \gamma(v)\Delta t_{I}(v),$$ which is the statement of time-dilation.

Some other notes:

• The result of Part 2 can be used to deduce equation $$(*)$$ in Part 1. This is the way the KT experiment implies $$(*)$$.
• The real experiment took place not only at different speeds, but also at different orientations.
• As I said in my post, the real experiments didn't involve precision timing with atomic clocks. So then what exactly was time-dilated? The answer is the light source. If you trace the logic carefully, you will find that the time-dilation only pertains to the light source. I invoked clocks explicitly only for the sake of exposition, but they are of course not needed here.