# Tagged Questions

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### Textbook disagreement on geodesic deviation on a 2-sphere [on hold]

Apologies if I have this completely wrong (and for the general long-windedness). I've searched online but can't find anything helpful/relevant. I'm trying to use the geodesic equation ...
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### Tricks for Computing Riemann Curvature Tensor with Levi-Civita connection

I am new to differential geometry, so far it seems to me that computing the Riemann tensor tends to be a rather tedious task, I wanted to know whether there are some tricks that I am missing. In ...
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### Length in polar coordinates

Say we are in 3 dimensions and use $(-++)$. If we have the metric $$ds^2=-dt^2+dr^2+r^2df^2(t),$$ then what is the third coordinate if the first two were $t$ and $r$? $$X^iX_i=-t^2+r^2+?$$
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### Static geodesics in GR

Can we find static geodesics of the type $$x^{\nu}=x_0^{\nu}+\delta_0^{\nu}\tau$$ in some space-time other than Minkowski's?
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### GR exercise: falling particles on earth's surface

I'm having some trouble with Exercise 5.1 in Shapiro's BH,WD&NS book, which goes as follows: Consider two particles of mass $m$ at distance $r$ and $r+h$, such that $h\ll r$, on the same ...
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### The commutator of Killing vectors

I'm going over an assignment for my general relativity course. My solution to the question below strikes me as too short, considering that it appeared in the "longer questions" section of the ...
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### Lagrangian for FRW metric

For the metric $$ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2),$$ $$L= \sqrt{-g_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}}$$ How does this become $$L= \sqrt{1-a^2 (\frac{dx}{dt})^2}~?$$ I guess ...
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### GPS Satellite - Special Relativity

I'm going through an old relativity assignment, and I've been asked to calculate the time dilation for a satellite which orbits the earth in 12 hours at 26000km from the surface, and travels at a ...
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### $c^4$ in Einstein field equations

I have read many derivations of Einstein field equations (done one myself), but none of them explain why the constant term should have a $c^4$ in the denominator. the $8{\pi}G$ term can be obtained ...
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### Conditions that the coordinate must satisfy in order to become local inertial

Consider the coordinate transformation $$\tilde x^a=x^a+\frac{1}{2}\Gamma^a_{bc}x^bx^c$$ I have shown that at the origin $O=(0,0,0,0)$, $$\frac{\partial\tilde g_{ab}}{\partial\tilde x^c}=0$$ ...
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### (Special Relativity) Points that can be seen by an observer

Let the metric be $$ds^2=(1+gz)^2dt^2-dx^2-dy^2-dz^2$$ where $g$ is a positive constant. Let an observer be stationary at $x=y=0$ on the surface $z=0$ and look upwards at an angle $\theta$, how ...
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### Wald problem 4 of chapter 4

I'm trying to derive equation 4.4.51 in Wald's GR book (the second order correction in $\gamma$ term for the Ricci tensor): where $g=\eta+\gamma$. So ...
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### Calculate the Riemann tensor and Ricci tensor [closed]

Given a metric tensor $\gamma_{ij}$ (where $i, j = 1, 2, 3$; the metric tensor of 3- dimensional space is denoted by $\gamma_{ij}$ to distinguish it from the metric tensor $g_{\mu\nu}$ of ...
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### Simple General Relativity Relation [closed]

Given the identity $$\nabla_a(R^{ab}-\frac{1}{2}R g^{ab})=0,$$ how do I then show that $R_{ab}=0$ implies $$\nabla_a R^a_{\space \space \space bcd}=0$$
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### Spacetime line elements and proper time (specific problem)

Sometimes, when we have a certain line element and we are given a worldline parametrised by a path parameter that is not necessarily proper time I don't completely understand some of the standard ...
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### Covariant Derivative with a Torsion Free Metric

Where $\triangledown$ is the covariant derivative and we are to assume that the connection is torsion free (that is, we can exchange the lower indices of the connection coefficients), how can I prove ...
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### 4 of Einstein equations without 2nd order time derivative

This question is related to my previous one and it was a homework problem and was due two weeks ago. Problem：prove that four of Einsteins' equations $$G_{0\nu} = 8\pi T_{0\nu}$$ have to 2nd order ...
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### causal sketches [closed]

I don't have much of an idea of how to draw causal sketches. I know that you need to work out the gradient of the light cones, which can be done using a given metric and using null vectors. But how do ...
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### Specify the Stress Energy Tensor and Calculate the Curvature

I have a simple question about general relativity and the Einstein field equations, I wonder if you can specify the stress energy tensor, i.e. specify some mass distribution in space and then ...
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### How to show that every Killing vector field is a matter collineation?

Various texts make this claim, but no proof is given. Explicitly, let $L$ denote the Lie derivative. Suppose $L_X g_{ab} = 0$ for some vector field $X$, called a Killing vector field. Suppose that ...
### In the static spacetime, the extrinsic curvature of hypersurface $t=constant$ is zero
How can I prove that in the static spacetime, the extrinsic curvature of hypersurface $t=constant$ is zero? My efforts all are failed. Any hint would be greatly appreciated.
Let's have metric $$ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - 2f(t - z, x, y)(dt - dz)^2.$$ I need to prove that it is an exact solution for Einstein equations in vacuum for \$\partial_{x}^{2}f + ...