wandering.the.cosmos
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I'm looking for recommendations for a good place that discusses at a basic level what physical length, time, simultaneity, etc. mean, for an arbitrary metric.
Landau does discuss this a bit, but in a way that confuses me -- for example he calls
[tex]\sqrt{g_{00}} dx^0[/tex]
"proper time", but in the SR limit this would just become dt, which isn't proper time, is it? Landau goes on to use this to derive distance in GR
[tex]\gamma_{ij} dx^i dx^j = \left( -g_{ij} + \frac{ g_{0i} g_{oj} }{g_{00}} \right) dx^i dx^j[/tex]
where my indices are spatial -- they run only from 1 to 3.
I'd like to understand where these expressions come from, and more importantly gain a good understanding of what length and time mean.
Landau does discuss this a bit, but in a way that confuses me -- for example he calls
[tex]\sqrt{g_{00}} dx^0[/tex]
"proper time", but in the SR limit this would just become dt, which isn't proper time, is it? Landau goes on to use this to derive distance in GR
[tex]\gamma_{ij} dx^i dx^j = \left( -g_{ij} + \frac{ g_{0i} g_{oj} }{g_{00}} \right) dx^i dx^j[/tex]
where my indices are spatial -- they run only from 1 to 3.
I'd like to understand where these expressions come from, and more importantly gain a good understanding of what length and time mean.