- #1
shahbaznihal
- 53
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I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field Newtonian metric tensor ##(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))##. I looked at the solution from a manual and it has the following equation,$$x^{\alpha '} = (\delta^\alpha_\beta + L^\alpha_\beta)x^\beta$$.
##L^\alpha_\beta## is a function of the Newtonian potential ##\phi##.
My question is: Is this a valid tensor equation?
The transformation is motivated by the idea that when ##\phi## is zero then you already have a locally inertial frame hence ##L^\alpha_\beta## are all zero and ##x^{\alpha '} = x^{\alpha}## which is very understandable. But is the equation ##x^{\alpha '} = (\delta^\alpha_\beta + L^\alpha_\beta)x^\beta## symbolically correct (because the superscripts don't balance out like they normally do in tensor calculus)? But may be if ##L^\alpha_\beta## is not a tensor then it does not have to obey those principles of tensor calculus.
##L^\alpha_\beta## is a function of the Newtonian potential ##\phi##.
My question is: Is this a valid tensor equation?
The transformation is motivated by the idea that when ##\phi## is zero then you already have a locally inertial frame hence ##L^\alpha_\beta## are all zero and ##x^{\alpha '} = x^{\alpha}## which is very understandable. But is the equation ##x^{\alpha '} = (\delta^\alpha_\beta + L^\alpha_\beta)x^\beta## symbolically correct (because the superscripts don't balance out like they normally do in tensor calculus)? But may be if ##L^\alpha_\beta## is not a tensor then it does not have to obey those principles of tensor calculus.