All Questions
Tagged with metric-tensor homework-and-exercises
453 questions
-2
votes
1
answer
68
views
Divergence of canonical energy-momentum tensor in QFT [closed]
I have to show that the divergence of the canonical energy-momentum tensor is zero, i.e. $$\partial^{\mu}T_{\mu\nu} = 0.$$
The Lagrangian is $$\mathcal{L} = \frac{1}{2}\partial^{\mu} \phi \partial_{\...
- 141
4
votes
1
answer
157
views
The variational derivative of the metric with respect to inverse metric
I have a somewhat naive question. However, consider the functional derivative
$$
\frac{\delta \partial_\mu g_{\alpha \beta}(x)}{\delta g^{\gamma \epsilon}(y)}.
$$
Is it should be
$$
\frac{\delta \...
- 85
0
votes
1
answer
134
views
Does this tensor identity hold in any kind of generality?
Assuming Minkowski spacetime, I am given an antisymmetric tensor $F^{\alpha\beta}$, and am asked to prove the following identity:
$$ F_{\mu}{}^{\alpha}{}_{,\nu}F^{\nu}{}_{\alpha} = -F_{\mu\alpha,\...
- 3,265
1
vote
0
answers
37
views
The contraction of Christoffel symbols [duplicate]
I have a question regarding the the contracted Christoffel symbols from David Tongs PDF on general relativity.
He wants to prove that
$$\Gamma^{\mu}_{\mu v}=\frac{1}{\sqrt{g}}\partial_v\sqrt{g}$$
...
- 19
1
vote
1
answer
88
views
Finding Killing vectors for hyperbolic space [closed]
I want to find the Killing vectors for the hyperbolic space, which is described by the metric
\begin{equation}
ds^2 = \frac{dx^2 + dy^2}{y^2}.
\end{equation}
I have found the Killing equations, which ...
- 131
0
votes
1
answer
68
views
How to obtain this relativistic limit for Rindler coordinates?
I have trouble following the calculation in this answer for Rindler coordinates. In particular, I am not sure how the second term is obtained in this equation:
$$f^x = m\frac{d^2x}{d\tau^2} + ma_0 \...
- 711
3
votes
0
answers
63
views
Components of Killing vectors being zero
The following is a question from a GR worksheet given by my professor:
Does the following line element:
$$ ds^2 = -dt^2 + e^{\alpha t}(dr^2+r^2\vartheta^2+r^2\sin^2\vartheta d\varphi^2), \qquad [\...
- 172
0
votes
0
answers
45
views
Equation of motion for $X^{\mu}$ (geodesic equation)
The action for a relativistic particle of mass $m$ in a curved $D$-dimensional is $$\tilde{S}_0=\frac{1}{2}\int d\tau (\dot{X}^2-m^2)$$ for particular gauge and $\dot{X}^2=g_{\mu\nu}(X)\dot{X}^{\mu}\...
- 634
1
vote
1
answer
52
views
How to calculate the functional derivative of a product?
Let $\omega_{J}^{I}=\omega^{I}_{\nu~J}dx^{\nu}$ be a one-form connection with values in the Lorentz group $SO(3,1)$ and $$B_{J}^{I}=B^{I}_{\mu\nu~J}dx^{\mu}\wedge dx^{\nu}$$ a two-form with values in ...
- 176
0
votes
0
answers
25
views
Finding coordinate transfromations by line element
I’m confused on how to generally approach these coordinate transformations: I initially thought we can set $dT^2=(dt-b/2dx)^2$ and $a^2dX^2= (a^2+\frac{b^2}{4})dx^2$. This way, when carrying out the ...
- 1
0
votes
0
answers
68
views
Extrinsic Curvature Calculation on the Sphere
Given the following 2+1 dimensional metric:
$$ds^{2}=2k\left(dr^{2}+\left(1-\frac{2\sin\left(\chi\right)\sin\left(\chi-\psi\right)}{\Delta}\right)d\theta^{2}\right)-\frac{2\cos\left(\chi\right)\cos\...
0
votes
2
answers
86
views
How to prove $ g^{\mu\nu}\Lambda^{\rho}{}_{\mu}\Lambda^{\sigma}{}_{\nu}=g^{\rho\sigma} $ for the inverse metric?
In Srednicki's book, we have
\begin{align*}
g_{\mu\nu}\Lambda^\mu{}_\rho\Lambda^\nu{}_\sigma=g_{\rho\sigma}
\end{align*}
and let $ \Lambda \to \Lambda^{-1} $, use the relationship $ (\Lambda^{-1})^\...
- 79
1
vote
0
answers
60
views
Are the quantity $\frac{\delta \sqrt{-g}}{\delta g_{\mu \nu}}$, $\frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} $ are computable? [duplicate]
From $\delta \sqrt{-g} = -\frac{1}{2} \sqrt{-g} g_{\alpha \beta}\delta g^{\alpha \beta} = \frac{1}{2}\sqrt{-g} g^{\alpha \beta} \delta g_{\alpha \beta}$, can we compute
\begin{align}
&\frac{\...
- 3,662
3
votes
1
answer
88
views
Circumference of ellipse in post-Newtonian metric
The post-Newtonian metric, in harmonic coordinates, is:
$$\tag{1}
\mathrm{d}s^2=-\left(1+\dfrac{2\phi}{c^2}\right)c^2\mathrm{d}t^2 + \left(1-\dfrac{2\phi}{c^2}\right)\mathrm{d}\mathbf{x}^2$$
where $\...
- 179
2
votes
1
answer
53
views
Does the van Stockum dust have closed timelike geodesics?
We work under this explicit form of the metric:
$$ds^2 = -dt^2 - 2br^2\,dtd\phi + (1-b^2r^2)r^2\,d\phi^2 + e^{-b^2r^2}(dz^2 + dr^2).$$
Since the metric coefficients only depend on $r$, we can ...
- 21
0
votes
1
answer
103
views
Raising and lowering indices of Kronecker delta [duplicate]
I have some confusion about how to raise the indices of the Kronecker delta.
To raise and lower indices we use the metric tensor, let's suppose to use the metric (+---).
I should have that $$g_{\mu\nu}...
- 91
0
votes
1
answer
93
views
Show that the dot product of two basis vectors in special relativity gives the metric
I'm reading Schutz's Introduction to GR and came across an exercise problem. The problem is the following:
Show that the vectors $\{\vec e_{\bar \alpha }\}$ obtained from
$$\vec e_{\bar \beta } = \...
- 55
1
vote
0
answers
78
views
Robertson-Walker metric exercise [closed]
I'm trying to solve an exercise from my astrophysics and cosmology class, the request is the following, starting from the RW metric expression:
$$ \begin{equation*}
ds^2=c^2 dt^2 - a^2 \left ( \frac{...
- 63
0
votes
1
answer
242
views
Calculating the determinant of the Kerr metric
I have been trying to calculate the determinant of the Kerr metric (described by equation 11.71, A First Course in General Relativity: 3rd Edition, B. Schutz):
$$\begin{align}ds^2 &=-\frac{(\Delta-...
0
votes
1
answer
392
views
Showing that derivative of energy-momentum tensor is equal to 0
Given,
\begin{equation}
T^{\mu\nu} = F^{\mu\lambda} F^\nu{}_{\lambda} - \frac{1}{4} \eta^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}.
\end{equation}
Here $(T^{\mu\nu})$ is the energy-momentum ...
- 111
-2
votes
1
answer
143
views
How do I compute the trace and the inverse of this tensor?
$$[X^{\mu\nu}] = [X] = \begin{pmatrix} 2 & 3 & 1 & 0 \\ 1 & 1 & 2 & 0 \\ 0 & 1 & 1 & 3 \\ 1 & 2 & 3 & 0 \end{pmatrix}$$
How to compute the trace of $X^{\...
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0
votes
1
answer
68
views
Understanding the derivation of Killing horizon surface gravity
In the book "A Relativist's Toolkit" by Eric Poisson, he explains surface gravity in section 5.2.4
The equation 5.40 says
$$ (-t^\mu t_\mu)_{;\alpha} = 2 \kappa t_\alpha \tag{5.40}$$
where $...
- 313
3
votes
2
answers
412
views
Self-studying GR. Stuck on Q3.20 in the 3rd edition of Schultz. Orthogonal coordinate transforms in Euclidian space
I am self-studying GR using "A first course in general relativity, 3rd edition". I'm doing my best to be diligent and work though the problems at the end of the chapter. But question 3.20 ...
- 33
2
votes
0
answers
72
views
Do two coordinate systems cover the same patch of the de Sitter manifold
I am self studying general relativity and there is some especially hard problem (it is called bonus problem in book) I am currently working on it, but I am trully stuck, so I would appreaciate all the ...
2
votes
0
answers
81
views
Proving that the Christoffel connection transforms like a connection
In Sean Carroll's intro to GR, he shows that a connection transforms as follows: $$\Gamma^{\upsilon'}_{\mu'\lambda'}=\frac{\partial x^\mu}{\partial x^{\mu'}}\frac{\partial x^\lambda}{\partial x^{\...
- 187
2
votes
0
answers
73
views
Asymptotically flat initial data sets for Schwarzschild [closed]
Wald Ch 11, problem 3 b reads:
"Show that every $t$ = constant hypersurface in the Schwarzschild solution is is an asymptotically flat initial data set."
The metric for Schwarzschild is of ...
- 61
3
votes
1
answer
80
views
Proportional null vectors [closed]
the past few days I've been studying special relativity and was just now making some exercices on it. One exercice was the following:
Let $U$ and $V$ be two null vector is a $d$-dimensional Minkowski ...
- 29
0
votes
1
answer
74
views
Tensor Index Manipulation
I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that
$$\partial_{\mu} ...
- 1
2
votes
0
answers
239
views
How does one justify Schwartz's answer to his problem 3.7 in "Quantum Field Theory and the Standard Model"?
In problem 3.7 of his textbook "Quantum Field Theory and the Standard Model", Schwartz gives a simplified Lagrangian density
$$
\mathcal{L}=- \frac{1}{2} h \Box h + \epsilon^a h^2 \Box h -\...
- 137
1
vote
1
answer
162
views
Star Radius in the Oppenheimer-Snyder metric using ADM formalism
I'm working with gravitational collapse models, in particular with the Oppenheimer-Snyder model.
Short list of the assumptions for those unfamiliar with the model, you have a spherical symmetric ...
- 1,854
1
vote
0
answers
94
views
Christoffel symbol and metric tensor [duplicate]
In B. Schutz's book A First Course in General Relativity, I read on the page 152:
$$ \Gamma^\alpha{}_{\mu \alpha} = \frac{1}{2} g^{\alpha \beta} g_{\alpha \beta, \mu} \tag{6.38} $$
Since $\left(g^{\...
- 21
0
votes
0
answers
261
views
Deriving the Ricci tensor on the flat FLRW metric
I am currently with a difficulty in deriving the space-space components of the Ricci tensor in the flat FLRW metric
$$ds^2 = -c^2dt^2 + a^2(t)[dx^2 + dy^2 + dz^2],$$ to find:
$$R_{ij} = \delta_{ij}[2\...
- 1
4
votes
1
answer
367
views
Coordinate derivation of non-metricity tensor
In the lecture of my course of Theories of gravity the definition of non-metricity tensor is:
$$Q(w,v,z)=-\nabla_{w}g(v,z)+g(\nabla_{w}v,z)+g(w,\nabla_{w}z).$$
First question: where could i find this ...
- 177
4
votes
2
answers
639
views
Confusion on metric determinant derivative
Maybe it is a stupid confusion. I need to compute the derivative of the metric determinant with respect to the metric itself, i.e., $\partial g/\partial g_{\mu\nu}$, but I have an indices confusion in ...
- 378
2
votes
2
answers
241
views
Calculating the Ricci tensor
I am currently working through an exercise to calculate the component $R_{22}$ of the Ricci tensor for the line element $ds^2=a^2dt^2 -a^2dx^2 - \frac{a^2e^{2x}}{2}dy^2 +2a^2e^xdydt -a^2dz^2$. The ...
- 99
2
votes
1
answer
290
views
How to obtain orthonormal tetrad basis for an infalling observer?
An eternal Schwarzschild spacetime in Painlevé-Gullstrand coordinates reads as
$ds^2 = -\left(1-\dfrac{2m}{r}\right)c^2~dT^2 + 2\sqrt{\dfrac{2m}{r}}c~dTdr + dr^2 + r^2\left(d\theta^2+\sin{^2\theta}~d\...
- 778
0
votes
4
answers
295
views
Why is $dt/d\tau=\gamma$? What is $dt/d\tau$ supposed to mean exactly?
I'm a math student trying to learn some physics by reading Susskind's The Theoretical Minimum. In the volume on special relativity he derives that $\frac{dt}{d\tau}=\gamma=1/\sqrt{1-v^2}$ and uses it ...
- 21
0
votes
1
answer
53
views
A question about $k_{\alpha \beta}:=\eta_{\mu \nu}L^{\mu}_{\alpha}L^{\nu}_{\beta}-\eta_{\alpha \beta}$
I have a question in the book "A First course in string theory" by Zwiebach. In page 20, the auther says $$k_{\alpha \beta}x^{\alpha}x^{\beta}=0\tag{2.42}$$ must hold for all values of the ...
- 167
0
votes
2
answers
113
views
Contravariant metric tensor with off-diagonal terms
I have to compute the contravariant metric tensor with off-diagonal terms, such as $g_{0i}\neq 0$. I started with the condition $$g_{\mu\nu}g^{\nu\lambda}=\delta^\lambda_\mu$$
but I don't know how to ...
- 338
1
vote
1
answer
258
views
The importance of metric signature in Ricci scalar
I have read this question Different signatures of the metric in Einstein field equations (and related posts) on the invariance of Einstein field equations under metric signature change.
However, there ...
- 179
-1
votes
1
answer
121
views
How to get rid of the affine parameter in geodesic equation?
I encounter a problem that require me to calculate the geodesic of
$$ds^2=\frac{dx^2+dz^2}{z^2}$$ with the endpoint $(x_L,0),(x_R,0)$. I get the answer $\ddot{x}-\frac{\dot{x}\dot{z}}{z}=0$ and $\...
- 133
-1
votes
2
answers
350
views
Construction of infinitesimal Lorentz transformation
I'm following the book from Greiner on relativistic QM and I got two questions here:
In (3.36b) where does the last expression come from? From which we get delta = delta + omega_nu^sigma + omega^...
- 45
1
vote
0
answers
49
views
Finding a closed timelike curve for a specific metric [closed]
I am tasked with finding a timelike curve in a specific metric that supports it. How can I approach this problem? Metric is $$ds^2=-dt^2+a(r) dr^2+ b(r) d\varphi^2 + c(r) dtd\varphi,$$ where $a(r)=(1+...
1
vote
2
answers
348
views
Metric tensor determinant under coordinate transformation
I've been studying GR through Wald's and Carroll's books, and I've been trying to derive one expression.
$$g(x^{\mu^\prime}) = \left|\dfrac{\partial x^{\mu^\prime}}{\partial x^{\mu}}\right|^{-2} g(x^\...
- 356
3
votes
1
answer
805
views
What is the Schwarzschild metric in cylindrical coordinates?
I was researching online for different metrics of spacetime out of curiosity, and I found one that was said to be Schwarzschild metric in cylindrical coordinates:
$$ds^2 = -\left(1-\frac{r_s}{r}\right)...
- 2,042
1
vote
1
answer
781
views
Scalar curvature of a 2-sphere via the Ricci tensor
Using the usual coordinates on a 2-sphere of radius $r$, I get the metric tensor $g_{\mu\nu}=\text{diag}(r^2, r^2\sin^2\theta)$ and so $g^{\mu\nu}=\text{diag}(1/r^2,1/r^2\sin^2\theta)$.
Hence the only ...
- 467
0
votes
0
answers
54
views
Tensor index of expression
When I do photon path integral quantization, I need to change variables like:
$$A^{\mu \prime}(x) \equiv A^{\mu }(x)-(\partial^2 g^{\mu \nu}-(1-\frac{1}{\xi}) \partial^{\mu}\partial^{\nu})^{-1} J_{\nu}...
- 1,461
1
vote
1
answer
103
views
String action in light-cone coordinates
I am going through textbook Einstein Gravity in a Nutshell by A. Zee and I got mathematically stuck at page 147 where he is talking about the classical string action using light cone coordinates. ...
1
vote
1
answer
125
views
Area of a sphere in curved 3D space
I'm having trouble finding any information on the derivation of the area of sphere in curved 3D space: $A = 4 \pi S^2_{\kappa}$, where $S_{\kappa} = R_o \sin(r/R_o)$.
How did it come about from $ds^2 =...
- 191
3
votes
1
answer
7k
views
Christoffel symbols for Schwarzschild metric
So, I am doing coursework for maths, and I'm trying to find out how I could calculate the Christoffel symbols for the Schwarzschild metric; however, I have realized that calculating them is too ...
- 53