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cianfa72

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- On the conditions to be able to build a synchronous reference frame in any spacetime

Hi,

reading the Landau book 'The Classical theory of Field - vol 2' a doubt arised to me about the definition of synchronous reference system (a.k.a. synchronous coordinate chart).

Consider a generic spacetime endowed with a metric ##g_{ab}## and take the (unique) covariant derivative operator ##\nabla_a## associated to it. From the Frobenius theorem we know that a timelike vector field ##\xi^a## is hypersurface orthogonal (w.rt. the given metric) if and only if the following holds (see for instance Wald appendix C):

$$\xi_{[a} \nabla_b \xi_{c]} = 0$$

So, given the metric and its associated covariant derivative operator ##\nabla_a## , we can solve the above equation for a timelike vector field ##\xi^a##. Now if a solution ##\xi^a## exists, then it is hypersurface orthogonal -- namely there is a scalar field, say ##t##, such that its ##t=const## level sets are spacelike hypersurfaces foliating the spacetime and orthogonal at each point to the vector field ##\xi^a## solution of the above equation.

If the above is correct, I do not fully understand the point made here wiki - synchronous coordinates -- the same as in Landau book - section 99. It seems that in

My concern is that the hypersurfaces we get using this procedure are

What do you think about ? Thank you.

reading the Landau book 'The Classical theory of Field - vol 2' a doubt arised to me about the definition of synchronous reference system (a.k.a. synchronous coordinate chart).

Consider a generic spacetime endowed with a metric ##g_{ab}## and take the (unique) covariant derivative operator ##\nabla_a## associated to it. From the Frobenius theorem we know that a timelike vector field ##\xi^a## is hypersurface orthogonal (w.rt. the given metric) if and only if the following holds (see for instance Wald appendix C):

$$\xi_{[a} \nabla_b \xi_{c]} = 0$$

So, given the metric and its associated covariant derivative operator ##\nabla_a## , we can solve the above equation for a timelike vector field ##\xi^a##. Now if a solution ##\xi^a## exists, then it is hypersurface orthogonal -- namely there is a scalar field, say ##t##, such that its ##t=const## level sets are spacelike hypersurfaces foliating the spacetime and orthogonal at each point to the vector field ##\xi^a## solution of the above equation.

If the above is correct, I do not fully understand the point made here wiki - synchronous coordinates -- the same as in Landau book - section 99. It seems that in

*any*spacetime we can always construct a synchronous reference frame starting from a given spacelike hypersurface. Namely take a family of timelike geodesics normal at each point to this spacelike hypersurface. Choose these lines as time coordinate lines and define the time coordinate ##t## as the length*##s##*of the geodesic measured starting from the given hypersurface; the result is a synchronous frame.My concern is that the hypersurfaces we get using this procedure are

*not*for sure orthogonal to the 4-velocity 's timelike geodesics. If we added the constrain that the timelike vector field ##\xi^a## was a timelike*Killing*vector field of the given spacetime then yes, moving along an isometry that would actually make sense.What do you think about ? Thank you.

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