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From: Sylvia Else <sylvia@email.invalid>
Newsgroups: sci.physics.relativity
Subject: Re: Proper time differences
Date: Tue, 9 Jul 2024 13:21:01 +0800
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On 08-July-24 10:45 pm, Stefan Ram wrote:
>    From various sources I gather,
> 
> dt = "gamma" d"tau".
> 
>    Where t is the coordinate time in the rest frame, "gamma"
>    is the Lorentz gamma factor and "tau" is the proper time.
> 
>    Now, if "gamma" is constant, I think we can replace the "d"
>    by "D" (triangle which is flat at its bottom), i.e., we can
>    use finite difference instead of infinitesimal ones:
> 
> Dt = "gamma" D"tau".
> 
>    I believe 0<="gamma"<=1, so, for an example, we can assume
>    "gamma" to be 0.5:
> 
> Dt = 0.5 D"tau",
> 
>    which means just,
> 
> D"tau" = 2 Dt.
> 
>    So, that would mean: For a moving thing the proper time
>    difference D"tau" (I assume: between two fixed events) is
>    /larger/ than the coordinate time difference.
> 
>    But since falling muons live longer, the proper time distance
>    should be /smaller/, not larger!
> 
>    What's wrong here? TIA!

"Time dilation" is a special case of the Lorentz transform, and due to 
continued lack of clarity on this point in popular science media, people 
tie themselves in knots by trying to use time dilation in situations 
that do not match the special case.

Apply the complete Lorentz transform to your problem, and any apparent 
contradictions will go away.

Sylvia.