Relativity of simultaneity

To illustrate the concept of relativity of simultaneity in special relativity, Einstein proposed a thought experiment where lightning strikes at the head and tail of a running train (points H and T, in short). There are two observers, Alice on the ground and Bob on the train exactly at the middle between H and T. The two events that lightning strikes H and T occur simultaneously to Alice, but not to Bob.

In the analysis of this thought experiment, it is important to distinguish the time that the observers receive the lights (received time) and the time that the light sent from both ends of the train (sent time). Both Newton and Einstein agree that Bob should receive the light from H earlier than the light from T since Bob is moving toward H and faraway from T. The disagreement is the sent time. Suppose in Alice's frame (ground), 
  • the train's length is $2L$;
  • the train's speed is $v$ and the speed of light is $c$;
  • Bob receives the light sent from H (T) at the time $t_H$ ($t_T$) assuming the lightning strikes at both H and simultaneously at $t=0$.
Then we have the relations $vt_H + ct_H = L$ and $ct_T = vt_T+L$, from which we can solve $t_H=\frac{L}{c+v}$ and $t_T=\frac{L}{c-v}$. Now what happens in Bob's frame (train)?

Newton's view

In Newton's world, there is absolute time and absolute space. So in Bob's frame, Bob still receives the light from H at the time $t'_H = t_H$ and receives the light from T at the time $t'_T = t_T$. Furthermore, in Bob's frame, both light travels the same distance from H and $T$ to Bob, which $L'=L$. 

However, the light's speed is different with respect to Bob: the speed of the light from H is $c+v$ while the speed of the light from $T$ is $c-v$. Therefore, it takes time durations $\frac{L'}{c+v}$ and $\frac{L'}{c-v}$ for lights travel from H and T to Bob.

As a result, in Bob's frame, the sent time from H and T can be deduced as $t'_H-\frac{L'}{c+v}$ and $t'_T-\frac{L'}{c-v}$, which are all equal to zero (simultaneity).

Einstein's view

In Einstein's view, the speed of light is the same $c$ in both Alice and Bob's frames. Therefore, in Bob's frame, the time durations from the lights to travel from H and T to Bob are all equal to $\frac{L'}{c}$. So the sent time is deduced as $t'_H-\frac{L'}{c}$ and $t'_T-\frac{L'}{c}$, which are not equal anymore!

For more quantitative calculations, we need to apply two relations from special relativity: one is time dilation and the other is length contraction (See the appendix of this post for the classical proof that can be understood by friends who are not in physics major): \begin{equation} t_H = \frac{t'_H}{\sqrt{1-\frac{v^2}{c^2}}}\,,\quad L=L'\sqrt{1-\frac{v^2}{c^2}}\,.\end{equation} The sent time from $H$ is actually \begin{equation} t'_H-\frac{L'}{c} = t_H \sqrt{1-\frac{v^2}{c^2}} - \frac{L}{c\sqrt{1-\frac{v^2}{c^2}}} = \frac{L}{c+v} \sqrt{1-\frac{v^2}{c^2}} - \frac{L}{c\sqrt{1-\frac{v^2}{c^2}}} = \frac{-\frac{v}{c^2}L}{\sqrt{1-\frac{v^2}{c^2}}}\,. \tag{1}\end{equation} Similarly, the sent time from $T$ can be calculated as \begin{equation} t'_T-\frac{L'}{c} = \frac{+\frac{v}{c^2}L}{\sqrt{1-\frac{v^2}{c^2}}}\,. \tag{2}\end{equation}

Lorentz Transformation

One can calculate the sent time from H in one line if knowing the Lorentz transformation \begin{equation} t'=\frac{t-\frac{v}{c^2}x}{\sqrt{1-\frac{v^2}{c^2}}}\,.\end{equation} The lightning strikes at H in Alice frame occurs at $t=0$ and $x=L$. Submitting $t=0$ and $x=L$ into the above equation immediately gives the same result as in Eq. (1).

Appendix

The following proofs does not relies on the Lorentz transformation. We only need to accept the fact that the speed of light is the same in all inertial frames, and be clear on how to measure the time and length in physics.

Time dilation

The time dilation $t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$ can be derived by considering a moving photon clock. Note that any periodic motion can used as a clock. The photon clock here is a light pulse bouncing vertically and periodically between two mirrors.

From the diagram, we have equations \begin{equation} t_0=\frac{2L_0}{c}\,,\quad t=\frac{2D}{c}\,, \quad D^2=L_0^2 + \left(v\frac{t}{2}\right)^2\,,\end{equation}
from which we can get the correct equation between $t$ and $t_0$.

Length contraction

Length contraction $L=L_0\sqrt{1-\frac{v^2}{c^2}}$ is usually proved by considering a clock travel from one place to another place on the earth. The key point is that the length between two places with respect to the earth is the proper length. 

Here, we provide an alternative proof that replies on the same photon clock used in proving the time dilation. This time we rotate the photo clock 90 degree so that the light pulse bounces horizontally along the moving direction. Suppose the light pulse leaves the mirror A at time $0$, reaches the mirror $B$ at time $t_1$ and comes back to the mirror A at the time $t$. From time dilation, we know that \begin{equation}  t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}\,,\quad t_0=\frac{2L_0}{c}\,.\end{equation} Since the light pulse travels along the same direction of clock motion, we have \begin{equation} c t_1 = L + vt_1\,,\quad c(t-t_1)+v(t-t_1)=L\,.\end{equation} With some algebra, we can get the correct relation between $L$ and $L_0$.

Location: 美国加利福尼亚帕罗奥图

Comments

Post a Comment

Popular posts from this blog

529 Plan

This post is for knowledge sharing only. It is not intended to be investment or tax advice. 529 Plans are tax-advantaged accounts specific for education saving: Contribution are made with after-tax dollars. Growth within the account is tax-free. Withdraw for qualified education expenses is also tax free. Roth IRA First 529 Plans are very similar to Roth IRA . But Roth IRA withdraw under  the 5-year rule  is tax free for any purpose. Therefore, unless for the state tax deduction in some states, it is always better to max out Roth IRA first (through backdoor and mega backdoor  for high income earners) before funding a 529 plan. Note: there is NO California state tax deduction for any 529 plan. 529 or Not? Here are some considerations in reality before funding a 529 plan: What if no qualified education expenses In the worst case that there will be no qualified education expenses, you can still take money out for non-qualified expenses, but you have to pay: federal income tax + 10% federa
Read more

How to offset W2 tax

This post is for knowledge sharing only. It is not intended to be tax or investment advice. 1. Contribute to pre-tax 401K and HSA As well known, pre-tax 401K and health saving account (HSA) are the only a few tax shelters for W2 workers.  2. Tax-loss harvesting If your total capital losses exceed your capital gains in a tax year, you can deduct up to \$3,000 (married filing jointly) from your ordinary income. If your losses are more than $3,000, you can carry over the remaining amount to future years, continuing to offset your income until all the losses are used up. A popular method to generate these deductible losses is  tax-loss harvesting . This involves selling a security at a loss to offset gains or reduce ordinary income, and optionally purchasing a similar, but not "substantially identical", security to avoid missing out on the market. However, be aware of the  wash-sale rule , which disallows claiming a loss if you buy the same or a substantially identical security
Read more

Retirement Accounts

Image
This post is for knowledge sharing only. It is not intended to be investment or tax advice. 0. Account Overview  Table 1 Common retirement  accounts in U.S. I summarizes the common retirement accounts in Table 1:  The second row in Table 1 lists the company-sponsored accounts that are offered through employers. The third row in Table 1 lists the individual retirement accounts (IRA) that is unrelated to employments. The columns in Table 1 are about different tax advantages as described below. WARNING: because of the tax advantages, there are also limitations and specific rules imposed by IRS for early withdrawal before the retirement age. For accounts listed in the second column,  Contributions are made with before-tax dollars . The money taken out will be taxed as regular income in the year of withdrawal. For accounts listed in the third column,  Contributions are made with after-tax dollars . The money taken out is  tax free . For after-tax 401K in the fourth column, Contributions are
Read more