Confusion about time dilation

  • I
  • Thread starter Haorong Wu
  • Start date
  • #1
188
33

Summary:

time dilation
Suppose there are two clocks. Clock B is moving with a speed of ##v## relative to clock A. Then clock A is moving with a speed of ##-v## relative to clock B.

Let ##t_0## be the time interval for two events in the framework of clock A, while ##t## be the time interval for the same two events in the framwork of clock B.

Then ##t=\gamma t_0##. Since ##\gamma## is greater than one, so for clock B time runs slower than for clock A.

But, also, ##t_0=\gamma t##, then time in clock A runs slower than in clock B.

This seems wrong. But where?

Thanks!
 

Answers and Replies

  • #2
29,953
6,350
Then ##t=\gamma t_0##.
This is not true in general. It is only true in the special case that the two events occur at the same location in frame A.
 
  • Like
Likes Haorong Wu and PeroK
  • #3
Mister T
Science Advisor
Gold Member
2,587
840
This seems wrong. But where?
They can't possibly be the same two events except in the trivial case where ##v=0##.
 
  • Like
Likes Haorong Wu
  • #4
Ibix
Science Advisor
Insights Author
6,829
5,676
As others have pointed out, ##t=\gamma t_0## is a special case that only applies when the events happen at the same spatial location. That will only apply to at most one frame (unless ##v=0##, as @Mister T notes).

The general expression is the Lorentz transforms:$$\begin{eqnarray*}
t'&=&\gamma\left(t-\frac v{c^2}x\right)\\
x'&=&\gamma\left(x-vt\right)
\end{eqnarray*}$$and their inverse:$$\begin{eqnarray*} t&=&\gamma\left(t'+\frac v{c^2}x'\right)\\ x&=&\gamma\left(x'+vt'\right) \end{eqnarray*}$$Note that you need to specify the time and position of each event and transform each one separately to get its transformed time and position. Then you can work out the time difference in either frame. (A useful trick is to declare that the origin is where and when one of the events is - ##(x,t)=(0,0)## transforms to ##(0,0)##.)

We generally recommend that you forget you ever heard of the time dilation formula and always use the Lorentz transforms until you are comfortable enough with them that you can see instantly whether you are allowed to use the time dilation formula in a given situation or not.
 
  • Like
Likes etotheipi, Dale and Haorong Wu
  • #5
188
33
Thanks, guys. I got it now. I forgot that the clock A remain still relative to the two events. Thanks again!
 
  • Like
Likes Dale
  • #6
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
14,350
6,724
I forgot that the clock A remain still relative to the two events.
Events are points in spacetime. To say that a clock is at rest relative to two events does not make sense. Instead, what you might mean is what @Dale said: that the two events occur at the same place in the rest frame of clock A.
 
  • #7
Erland
Science Advisor
738
136
  • Like
Likes vanhees71

Related Threads on Confusion about time dilation

  • Last Post
Replies
15
Views
2K
  • Last Post
Replies
22
Views
7K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
10
Views
1K
Replies
10
Views
4K
Replies
10
Views
2K
Replies
12
Views
2K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
14
Views
1K
Top