All Questions
48 questions
2
votes
1
answer
129
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Independent Components of the Riemann Curvature Tensor
I am struggling to understand a general method to calculate the independent components of the Riemann Curvature Tensor (RCT).
Firstly, as far as I am aware the number of independent components of the ...
- 155
1
vote
1
answer
64
views
Verifying whether the vanishing term in the Quasi-minkowskian metric is indeed a tensor
In Weinberg's Gravitation and Cosmology, on page 165 he notes that $h_{\mu \nu}$ is lowered and raised with the $\eta$'s since unlike $R_{\mu\kappa}$ it is not a true tensor (or at least implies it). ...
- 11
0
votes
0
answers
56
views
Is the Bianchi-identity conformally invariant?
I am trying to show that for a conformal transformation $\tilde{g}_{ab} = \Omega^2 g_{ab}$ the divergence of the non-physical Einstein-tensor $\tilde{G}_{ab}$ (i.e. the Einstein tensor corresponding ...
8
votes
3
answers
806
views
Why does the Weyl tensor not show up in the Einstein field equations?
In the Einstein field equations, the only tensor that shows up is the Ricci tensor and the metric tensor, together with the Ricci scalar. The Weyl tensor though is a tensor that is a part of the ...
- 460
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
7
votes
4
answers
381
views
Can you get $g_{\mu\nu}$ from $R^\lambda_{\alpha\beta\gamma}$?
From the Christoffel symbols it is easy (although cumbersome) to get the Riemann tensor $R^\lambda_{\alpha\beta\gamma}$ from the metric tensor $g_{\mu\nu}$. But is it possible to reverse the procedure?...
- 751
1
vote
1
answer
112
views
Difference between $R^{a}_{bcd}$ and $R_{abcd}$ Riemann tensor types
What is the intuitive, geometrical meaning regarding the usual mixed Riemann tensor $R^{a}{}_{bcd}$ with respect to its purely covariant counterpart $R_{abcd}$?
- 751
3
votes
0
answers
147
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Physical meaning of the Riemann curvature tensor with all 4 lower indexes
Since Riemann with 1 upper and 3 lower index mean the changes of a vector after being parallel transported around a closed loop, does Riemann with all 4 lower indexes mean the changes of a co-vector ...
- 1,248
0
votes
0
answers
129
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What is the connection between the Ricci tensor and the metric flatness?
Actually, this question was answered by Lawrence B. Crowell,
but I would like to explore this topic further.
Can anyone give me please references on where I can find it?
5
votes
1
answer
1k
views
Confusion regarding Riemann Tensor and Ricci Tensor
Ricci Tensor is the contraction of the Riemann Tensor. Even if all the components of the Ricci Tensor is zero, that doesn't mean that the spacetime is flat. If all the components of the Riemann Tensor ...
user355398
7
votes
1
answer
547
views
Is there are relationship between the Ricci scalar and the determinant?
On the one hand the determinant (technically the absolute value of the determinant) represents the "volume distortion" experienced by a region after being transformed.
On the other hand the ...
- 1,558
4
votes
1
answer
271
views
Torsion-free and compatible connection for non-symmetric non-degenerate $(0,2)$ tensor field?
In order to find a manifestly unique expression for the connection coefficients in terms of a given (possibly non-symmetric) tensor $g_{ab}$ usually two additional properties are assumed:
Torsion-...
- 467
1
vote
1
answer
768
views
Computing the components of the Ricci scalar
After contracting the Riemann tensor to the Ricci tensor, I have a 2 times covariant tensor. Thus, before contracting this to the Ricci scalar, I need to use the metric tensor to transform it into a $(...
user326230
0
votes
1
answer
144
views
Section 5.1 of Wald's *General Relativity* - part II
I am asking this in continuation of this question.
From the answer of that link and from section[5.1] of 'Homogeneity and Isotropy' from General Relativity by Robert M. Wald (pages 91-92, edition 1984)...
- 117
1
vote
0
answers
266
views
How to compute the scalar $^{(4)}R_{\mu\nu} \; ^{(4)}R^{\mu\nu}$ in the ADM formalism in General Relativity?
In the Arnowitt-Deser-Misner (ADM) formalism in General Relativity, the line element takes the form
$$
ds^2 = - N^2 dt^2 + \gamma_{ij} ( N^i dt + dx^i) (N^j dt + dx^j) \ ,
$$
where $\gamma^{ij}$ is ...
- 3,764
1
vote
1
answer
310
views
Issue calculating the scalar curvature for static, spherically symmetric spacetime
I was trying to follow the Cartan method to find the curvature as outlined here, in the case of a static, spherically symmetric spacetime. It all seemed to work fine, and for a metric of
$$ds^2 = -a^...
- 2,587
4
votes
1
answer
292
views
Riemann curvature tensor is the only tensor that you can write down that has two derivatives of the metric tensor
Is the statement in the main question correct? Can someone send me a proof (link, pdf file etc.)
- 143
1
vote
1
answer
246
views
Intuition for the indices of the Riemann tensor
I have some trivial questions about the Riemann tensor.
$R_{\alpha\beta\mu\nu}$, is the first two indices reserved for the components of the geometric object and the last two for the path it travels ...
- 117
1
vote
0
answers
191
views
Carroll's GR: The Schwarzschild Metric
On page 195 of Sean M Carroll's An Introduction to Special Relativity: Spacetime and Geometry, he calculates the Christoffels and Riemann tensors for a generic metric that is static and has spherical ...
- 589
2
votes
2
answers
355
views
What can be derived from the metric tensor? [closed]
I am working on a computational project about General Relativity. In this process, I want to code 'the stuff' that can be derivable from the metric tensor. So far, I have coded Riemann Tensor, Weyl ...
- 3,160
0
votes
0
answers
68
views
Two Definitions of the Weyl Tensor
I'm reading "Textbook in Tensor Calculus and Differential Geometry" by Prasun Kumar Nayak and came across the Weyl tensor/projective curvature tensor $C_{kijl}$. The book states that
$$C_{...
- 1,047
2
votes
1
answer
123
views
Different Ricci tensors for the same metric?
Today I was reading Carrol's book on General Relativity and got a bit confused. In the book, we are given the following metric $$ds^2 = - e^{-2 U(t,r)} dt^2 + e^{2 V(t,r)} dr^2 + r^2 d\Omega^2$$ from ...
- 1,327
2
votes
1
answer
619
views
Einstein field equation in vacuum : why $G_{\mu\nu} = 0 \implies R_{\mu\nu} = 0$
Starting from the Einstein field equation (without the cosmological constant),
$$
\underbrace{R_{\mu\nu} - \frac{1}{2}g_{\mu\nu} R}_{G_{\mu\nu}} = \underbrace{\frac{8\pi G}{c^4}}_{\kappa} T_{\mu\nu}
$$...
- 307
1
vote
1
answer
326
views
Proof that the Einstein Tensor is the Contraction of the Double Dual of Riemann
How can one prove that the Einstein tensor (as it is usually defined in the field equations) is the contraction of the double of the Riemann curvature tensor?
Specifically, I want to show
$$
R^\mu_\nu-...
- 13
0
votes
0
answers
322
views
Riemann tensor in linearized gravity
Assuming that $g_{\mu \nu}(x) = η_{\mu \nu} + h_{\mu \nu}(x)$, determine the first-order form of $R_{\lambda \mu \nu}^{\rho}$ in $h_{\mu \nu}$. The resulting theory is known as linearized gravity.
...
0
votes
0
answers
88
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Proving $R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}=\cfrac{48G^2M^2}{r^6}$
Show that in the case of the Schwarzschild metric: $R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}=\cfrac{48G^2M^2}{r^6}$
I am currently studying Relativity on my own. But in my book I found that this ...
- 65
0
votes
1
answer
103
views
Is it possible to write the Hilbert Action as a product of two identical tensors?
We know the Maxwell action can be written as the tensor product of the tensor $F^{ab}$ with itself. $F^{ab}F_{ab}$
[Edit: This bit I forgot to mention in the original quesiton]
Using the product rule ...
user84158
1
vote
1
answer
952
views
Covariant Differentiation on a Riemann Tensor
Working on an assignment I came across the following problem:
Show explicitly the Bianchi identity:
$$ R^{a}_{b[cd;e]} = 0 $$
where ; denotes covariant differentiation
$$
R^{a}_{bcd} = \frac{1}{2}g^{...
- 348
-1
votes
1
answer
44
views
What is the implication if the Kretschmann scalar for two metrics are same? [closed]
Does it mean that they are describing the same spacetime?
- 97
0
votes
1
answer
116
views
How to interprete this singularity? [closed]
I am calculating the Kretschmann scalar for the Schwartzchild metric. This is the graphic I get:
Where $R$ is the radial coordinate and $x=\cos(\theta)$.
So, there is the singularity at $R=0$ as it ...
- 1,187
1
vote
1
answer
245
views
Do the Christoffel symbols $\Gamma_{rn}^w\partial_sV_w = \Gamma_{sn}^w\partial_rV_w$?
In lecture 3 (about 97 min into the lecture) of Leonard Susskind's general relativity course, he suggests finding the Riemann curvature tensor in terms of the Christoffel symbols as an exercise.
I ...
- 53.7k
0
votes
0
answers
162
views
Ricci Curvature Tensor in a static gravitational field (non-relativistic)
Pg 171 of "Tensors, Relativity and Cosmology"
The non-relativistic limit of the metric in a static gravitational field is defined as $$ds^2=\left(1+\frac{2 \phi}{c^2}\right)(dx^0)^2+g_{\alpha \beta}...
- 1,047
1
vote
1
answer
507
views
Lowering index of Riemann tensor
I'm trying to undertand the lowering of index of Riemann curvature tensor, but I'm not sure what I have to do.
I know that $R_{ebcd} = g_{ea}{R^a}_{bcd}$. But let's say I have the coordinates ($t,r,\...
1
vote
1
answer
214
views
Proof of Schur's Theorem
On Pg. 123 of Schaum's Tensor Calculus:
At an isotropic point of $R^n$ the Riemannian curvature is given by $$K=\frac{R_{abcd}}{g_{ac}g_{bd}-g_{ad}g_{bc}}=\frac{R_{abcd}}{G_{abcd}}$$
for any ...
- 1,047
0
votes
1
answer
1k
views
Proof of first Bianchi identity
The proof is often simplified by using the following theorem:
"If the metric tensor $(g_{ij})$ is positive definite, then, at the origin of a Riemannian coordinate system $(y^i)$, all $\partial g_{...
- 1,047
1
vote
2
answers
1k
views
Indices of the Riemann Tensor of the first kind
When establishing the identity $V^i_{,kl}-V^i_{,lk}=-R^i_{tkl}V^t$ (, denotes covariant differentiation), one of the steps involves raising one of the indices of the Riemann Tensor of the first kind . ...
- 1,047
0
votes
2
answers
2k
views
Proof that the scalar curvature of a two-dimensional space can be expressed by only one component of the Riemann tensor
So I'm working on a question that asks for a proof that in two-dimensional space, the scalar curvature is given by:
$$R = \frac{2R^1{}_{212}}{{g}_{22}}$$
Now, I've been playing around with the ...
- 163
1
vote
1
answer
123
views
A question about the expression of Riemann tensor in Landau & Lifshitz
I was reading Landau & Lifshitz "The Classical Theory of Fields" and there is a expression at the beginning of section 92-Properties of the curvature tensor I don't understand. The author ...
- 11
3
votes
3
answers
348
views
Doubt about the vacua equations of General Relativity
I'm facing a quite annoying conceptual problem concerning the Einstein Field Equations (EFE) in so called "vacuum". This problem is both physical and mathematical.
So, in a elementary point of view, ...
- 3,159
10
votes
1
answer
8k
views
Why is Minkowski spacetime in polar coordinates treated in texts as flat spacetime?
Taking 3-D Minkowski spacetime line element in General Relativity:
$$ds^2=-c^2dt^2+dx^2+dy^2+dz^2, $$
when considering a change into spherical coordinates leads to:
$$ds^2=-c^2dt^2+dr^2+r^2\left(d\...
- 392
1
vote
1
answer
316
views
Einstein notation and conventions when raising/lowering indices with the metric
I was trying to find components of the Riemann tensor and it occurred to me that there could be an issue with my notation. For example, if one particular component of the tensor is
$$
R^{\...
- 63
7
votes
4
answers
2k
views
On the uniqueness of the Riemann-Christoffel tensor
According to Section 6.2, Gravitation and Cosmology by Weinberg, the Riemann-Christoffel tensor is the only tensor that can be constructed out of the second (or lower) order derivatives of the metric ...
user87745
4
votes
1
answer
1k
views
Flat 3D space described with spherical coordinates VS curved space being the surface of a sphere
I would like to ask if there is a way to know how to find out if a space is flat or curved given a metric that could describe a flat space in curvilinear coordinates or just curved space.
For ...
user99651
4
votes
1
answer
4k
views
What exactly does the Kretschmann scalar implies and how does it work?
From the General Relativity class lectures I understood that this particular invariant, the Kretschmann scalar namely
$$R_{\mu\nu\lambda\rho} R^{\mu\nu\lambda\rho}$$
is really important because, ...
- 3,735
0
votes
1
answer
94
views
Show that $R_{\mu\nu}=C g_{\mu\nu}$ from the vacuum Einstein equation with a nonzero $\Lambda$ [closed]
If I begin with the vacuum field equation with a nonzero cosmological constant:
$$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=0$$
How can I show that
$$R_{\mu\nu}= \frac{\Lambda}{\frac{D}{...
- 11
0
votes
1
answer
459
views
Weyl scalar calculation
I'm trying to compute Weyl scalars, but don't really understand the formulae for them, in the sense I don't understand how to compute them. Let's take
$\Psi_{2}=C_{1342}=C_{pqrs}l^{p}m^{q}\bar{m}^{r}...
- 427
-1
votes
1
answer
459
views
Metric tensor in General Relativity or otherwise [closed]
What is the metric tensor?
How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices?
How it relates to distance function (metric) and angles?
How does ...
- 19
15
votes
1
answer
1k
views
What are the local covariant tensors one can form from the metric?
Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain ...
user4552