What is the physical significance of locally Minkowskian but not necessarily geodesic coordinates?
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Consider an event PP in spacetime and a coordinate system xμxμ such that at PP:
gμν=ημνgμν=ημν
gμν,σ=0gμν,σ=0, or equivalently the Christoffel symbols vanish at PP.
xμxμ thus constitutes a locally Minkowskian geodesic coordinate system at PP and hence represents a local inertial frame (LIF). We can then write the metric at PP as ds2=dx21−dx22−dx23−dx24ds2=dx12−dx22−dx32−dx42 and read off immediately the physical signficance of the coordinates: dx1dx1 is the time as measured by this LIF and dx2,dx3dx2,dx3 and dx4dx4 are ruler distances along three orthogonal space directions in this LIF.
My question is: Consider a different coordinate system xμ′xμ′ (using the primed index notation) such that condition 1. holds but condition 2. is not known to hold*. We can still write the metric at PPin the form ds2=dx21′−dx22′−dx23′−dx24′ds2=dx1′2−dx2′2−dx3′2−dx4′2, but what physical signficance, if any, do the dxμ′dxμ′have now?
*I believe this is the case whenever one encounters a general diagonal metric and tries to interpret its coordinates by 'locally scaling' the coordinates so that the metric at a particular event is ημνημν.
gμν=ημνgμν=ημν
gμν,σ=0gμν,σ=0, or equivalently the Christoffel symbols vanish at PP.
xμxμ thus constitutes a locally Minkowskian geodesic coordinate system at PP and hence represents a local inertial frame (LIF). We can then write the metric at PP as ds2=dx21−dx22−dx23−dx24ds2=dx12−dx22−dx32−dx42 and read off immediately the physical signficance of the coordinates: dx1dx1 is the time as measured by this LIF and dx2,dx3dx2,dx3 and dx4dx4 are ruler distances along three orthogonal space directions in this LIF.
My question is: Consider a different coordinate system xμ′xμ′ (using the primed index notation) such that condition 1. holds but condition 2. is not known to hold*. We can still write the metric at PPin the form ds2=dx21′−dx22′−dx23′−dx24′ds2=dx1′2−dx2′2−dx3′2−dx4′2, but what physical signficance, if any, do the dxμ′dxμ′have now?
*I believe this is the case whenever one encounters a general diagonal metric and tries to interpret its coordinates by 'locally scaling' the coordinates so that the metric at a particular event is ημνημν.
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✔️✔️Consider a different coordinate system xμ′xμ′ (using the primed index notation) such that condition
1. holds but condition
2. is not known to hold*.
We can still write the metric at PPin the form
ds2=dx21′−dx22′−dx23′
−dx24′ds2=dx1′2−dx2′2−dx3′2−dx4′2, but what physical signficance, if any, do the dxμ′dxμ′have
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