Spacetime interval and the metrric

  • #1
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This may seem an odd question but it will clear something up for me. Are "The spacetime interval is invariant." and the "The spacetime metric is a tensor." exactly equivalent statements? Does one imply more or less information than the other?

Thanks!
 

Answers and Replies

  • #2
Matterwave
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They aren't exactly equivalent, but they are closely related. The space time interval is ##ds^2=g_{ab}dx^a dx^b##. Because the metric ##g_{ab}## is a tensor, contracting it with (infinitesimal) vectors ##dx^{a}## will yield an invariant scalar ##ds^2##.
 
  • #3
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They aren't exactly equivalent, but they are closely related. The space time interval is ##ds^2=g_{ab}dx^a dx^b##. Because the metric ##g_{ab}## is a tensor, contracting it with (infinitesimal) vectors ##dx^{a}## will yield an invariant scalar ##ds^2##.
Thanks! A little confused why you referred to ##ds^2## as a scalar. Isn't it a four-vector?
 
  • #4
bapowell
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Thanks! A little confused why you referred to ##ds^2## as a scalar. Isn't it a four-vector?
Nope. It's the scalar product of (differential) four-vectors.
 
  • #5
Matterwave
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Thanks! A little confused why you referred to ##ds^2## as a scalar. Isn't it a four-vector?

Why would it be a 4 vector? What are the 4 components that you are thinking of? It is a scalar because it's just 1 single number, which does not change under an arbitrary Lorentz transformation.
 
  • #6
19
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Got it now. Thanks very much.
 

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