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Trysse
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- TL;DR Summary
- Proper time along a light-like curve isn’t undefined; it’s zero. The infinities arise only from
for , not from the geometry itself. Using avoids these issues.
I’ve been following this recent discussion on the speed of light and its implications in special relativity, and it got me thinking about proper time along light-like curves and the use of the Lorentz factor, .
One point that stood out to me is the claim that proper time along a light-like curve is undefined because becomes infinite when . While this is technically correct for , the apparent “undefined” nature of proper time comes from ’s structure, not from any fundamental issue with proper time itself.
Proper time is often expressed as
it can also be written as
If we focus directly on rather than , the situation becomes much simpler. For light-like motion ( ), , and so the proper time is exactly zero. This result aligns perfectly with the idea that no proper time elapses along a light-like curve, such as the path of an electromagnetic signal in Minkowski spacetime.
The infinities associated with as are thus artifacts of its structure, not a reflection of the spacetime geometry itself. By focusing on , we avoid these complications entirely and gain a clearer understanding of proper time along light-like paths.
One point that stood out to me is the claim that proper time along a light-like curve is undefined because
Proper time is often expressed as
it can also be written as
If we focus directly on
The infinities associated with
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