General and Special Relativity Minkowski spaces

[kikitard]

Homework Statement

In attached image

Homework Equations

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The Attempt at a Solution

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A start would be fantastic!

 

[Dick]

If you don't have ANY idea how to start ANY of these you might be in the wrong course. 3(a) just wants you describe the metric as a matrix. If (dt,dx,dy) is the vector $dx^\mu$ what matrix $g_{\mu \nu}$ makes $dx^\mu dx^\nu g_{\mu \nu}$ equal to the metric expression?

 

[kikitard]

I'm pretty sure I am in the wrong course, but it is required, which is why I've turned to the internet for help.

Am I correct in saying that the dxν = transpose of dxμ, and so gμν is the matrix with row vectors (-1,0,0) (0,1,0) (0,0,1), or am I off base?

 

[Dick]

I'm pretty sure I am in the wrong course, but it is required, which is why I've turned to the internet for help.

Am I correct in saying that the dxν = transpose of dxμ, and so gμν is the matrix with row vectors (-1,0,0) (0,1,0) (0,0,1), or am I off base?

That's a good start. Absolutely right. Now try 3(b).

 

[kikitard]

Using this formula (attached) can i retain the β in the formula, giving 1/2g$\mu\beta((∂gβ\alpha/∂xβ)+(∂gββ/∂x\alpha)-(∂g\alpha\beta/∂xβ))$

we know gab=gba so the first and third terms in the brackets cancel
giving 1/2g$\mu\beta(∂gββ/∂x\alpha)$
?
Would the partial derivative wrt xalpha be 0, as there are no xalpha terms contained in gββ?

 

[Dick]

Using this formula (attached) can i retain the β in the formula, giving 1/2g$\mu\beta((∂gβ\alpha/∂xβ)+(∂gββ/∂x\alpha)-(∂g\alpha\beta/∂xβ))$

we know gab=gba so the first and third terms in the brackets cancel
giving 1/2g$\mu\beta(∂gββ/∂x\alpha)$
?
Would the partial derivative wrt xalpha be 0, as there are no xalpha terms contained in gββ?

That's hard to read. But all of the metric components are constant. So all of the partial derivatives of the metric are zero. So the Christoffel symbols are?

 

[kikitard]

That's hard to read. But all of the metric components are constant. So all of the partial derivatives of the metric are zero. So the Christoffel symbols are?

Ahh that makes sense, all partial derivatives are zero, so the Christoffel is also zero.

for c) I managed to get the attached image as the geodesic with affine λ
in d) i must solve this (presuming i did it correctly), could you point me towards the right method?

 

[DimReg]

for c) You derived the geodesic equation in general it looks like. You got the right answer for that, but you can also try to find the specific case geodesic equation.

Regardless, in part b) you found that the christoffel symbol was zero. What does that imply about your answer to part c)? Does that give you something easy to solve for part d)?