What's incorrect with using the two formulas for Lorentz contraction and time dilation at the same time?

Question

Lorentz contraction: $L = L_0 / \gamma$

Time dilation: $t = \tau \gamma$

Using the above two formulas at the same time leads to contradiction to the principle of the constancy of light velocity.

What’s wrong with using the above two formulas at the same time?

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What I understood about this question is as follows.

1. The length of the moving body in the formula of Lorentz contraction is measured in the stationary system of the observer.

2. The time interval of the moving body in the formula of time dilation is measured in the stationary system of the moving body.

Thus, these two formulae are not symmetrical against the exchange of time and space. These two formulas are symmetrical only against the exchange of time and space together with the exchange of coordinate systems used to measure time/space intervals.

Answer 1

The two different formulas are based on different assumptions.

The time dilation formula assumes that there are two events in spacetime and that the two events are in the same position in one reference frame. The length contraction formula assumes that there are two worldlines in spacetime and that the two worldlines are at rest in one reference frame.

For light (null geodesic), neither of these sets of assumptions can apply. So you cannot use either of these formulas for light, and certainly not both together.

The formula that works in general is the Lorentz transform. I recommend to beginning students of relativity that they not use the simplified length contraction and time dilation formulas. Just use the Lorentz transform. It will automatically simplify to the time dilation and length contraction formulas when appropriate, but you will avoid situations like this that arise from incorrectly using the simplified formulas when the assumptions are violated.

Answer 2

From this and your previous question, I suspect your confusion stems from the interpretation of $L$ in the length contraction formula. In fact this is something that confused me a lot when I was starting out.

Consider two observers attached to frames $S$ and $S^\prime$, with $S^\prime$ moving at speed $v$ relative to $S$ in the $x$-direction. Let their coordinates coincide at the origin. When we derive the formula for time dilation, we consider a change in time of $\Delta t=t_2-t_1$ in the $S$ frame. Performing a Lorentz transformation then gives a that same change in time in the $S^\prime$ frame as $\Delta t^\prime=t_2^\prime-t_1^\prime=\gamma(t_2-t_1)=\gamma\Delta t$ since $x=0$ for the observer in $S$. So we arrive at the familiar $$\Delta t^\prime=\gamma\Delta t,\tag*{(1)}$$ which tells us that the time for the observer in $S$ to reach $t_2$ is seen to be dilated to the observer in $S^\prime$.

Naively, we might try next to do the same thing for the spatial coordinate(s). Say we consider a length $\Delta x=x_2-x_1$ at $t_1=t_2=0$. Going through the same motions, we find that $$\Delta x^\prime=\gamma\Delta x.\tag*{(2)}$$ But wait. This isn't the correct form of the length contraction formula. It should really be $\Delta x^\prime=\Delta x/\gamma$. What gives?

The way to understand this is to realize that $L$ is supposed to be a length. Equation 2 is the $x^\prime$ distance between two points (the ends of the rod, say) at different times. This is obviously no good. While for the derivation of the time dilation formula it was okay to compare the intial and final times despite the fact that each observer had changed spatial position from the perspective of the other, the same is not okay for measuring the length of a rod. You need to measure the spatial positions of either end at the same $t^\prime$ coordinate.

This is something that can be made much clearer by a spacetime diagram:

(R.L. Herman, 2008)

Considering the diagram on the right, the $x^\prime$ distance between the pair of diagonal dashed lines is the $\Delta x^\prime$ in eq. 2. The contracted length however is the $\Delta x^\prime$ shown in the diagram, which is a distance between points at simultaneous $t^\prime$.

Answer 3

It may help to look at a case where both length contraction and time dilation are relevant in analyzing the motion of light, and how the formulas are consistent when correctly applied.

Consider a "light clock", a channel in which a light pulse reflects back and forth. In its rest frame, the channel has length $0.5$ and so light takes time $1$ to complete one period (back and forth), in units where $c = 1$.

Now, what is the clock's period in a frame where the clock moves at speed $v$?

Typically this is analyzed with motion perpendicular to the channel, but that requires at least two spatial dimensions. If the clock's motion is parallel to the channel, we can limit the problem to one dimension, and have the "bonus" of bringing in the length contraction formula.

The contracted length of the channel is $0.5/\gamma$, where $\gamma = (1 - v^2)^{-1/2}$. When the light is going rightward, the apparent relative speed between the light and the channel in this frame is $1 - v$, and when the light is going leftward, this apparent speed is $1 + v$. (Ordinary addition applies because we are simply doing kinematics in this one reference frame.)

So the total time to propagate from one side to the other and back is $$t = \frac{0.5/\gamma}{1 - v} + \frac{0.5/\gamma}{1 + v} = \frac{1/\gamma}{1 - v^2} = \gamma.$$ This is dilated from the rest period of $1$ exactly as expected.

This represents a valid use of length contraction and time dilation in evaluating the behavior of light.