Time Dilation: Self-Consistency Problem Explained

[mathman44]

Hi all.

I'm having trouble getting an intuitive understanding for the following situation. Let frame A and frame B be moving with relative velocity v.

It's true that a clock in frame A will be time dilated with respect to a clock in frame B, but also that a clock in frame B will be time dilated with respect to a clock in frame A.

i.e. any time between two events measured in frame A must be multiplied by gamma to get the corresponding time measured between those two events in frame B, and correspondingly, that time as measured in frame B must be multiplied by gamma to get the corresponding time as measured in frame A. Obviously this would mean that

$$t_a = \gamma^2{t_a}$$

Can anyone explain this apparent contradiction?

 

[Doc Al]

i.e. any time between two events measured in frame A must be multiplied by gamma to get the corresponding time measured between those two events in frame B,

In general, that's not true. You need the full Lorentz transformations to translate the time between two arbitrary events from one frame to another.

In the special case of events that take place at the same location in A, which could be measured with a single collocated clock in A, then the simple time dilation formula would apply. (Note that events that take place at the same location in A must of necessity take place at different locations in B; so it's true that Δt_B = γΔt_A, but Δt_A ≠ γΔt_B.)

 

[mathman44]

Ah... yes, if the events occur at one location in space in one frame, they must be separated in space in the other frame.

Cheers!

 

[ghwellsjr]

Mathman, now that Doc Al has addressed the second part of your post, can you explain how that impacts the earlier part where you said:
"It's true that a clock in frame A will be time dilated with respect to a clock in frame B, but also that a clock in frame B will be time dilated with respect to a clock in frame A."

Specifically, does his answer mean that one or both of your "true" statements are not true?

It appears to me that he was focusing on just your first statement and it appears that what you call "a clock in frame A" is what he is calling "a single collocated clock in A" but I'm wondering that if you still consider this to be a true statement, where is the clock in frame B or what clock in frame B were you talking about?