All Questions
Tagged with metric-tensor curvature
364 questions
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How to do expansion of Lagrangians in terms of parametrized metric
For a parametrized metric $$g_{00}= -e^{2\varphi}, g_{0i}=e^{2\varphi}A_{i}, g_{i0}=e^{2\varphi}A_{j}, g_{ij}=e^{-2\varphi}(\delta_{ij}+\sigma_{ij})-A_{i}A_{j}.$$
How to expand $
\sqrt{-g} R(g_{\mu\nu}...
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69
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Error in Di Francesco et al "Conformal Field theory" Eq 9.119?
In deriving equation 9.119 in their book the authors appear to claim that the metric variation of the Ricci tensor obeys
$$
g^{\mu\nu} \delta R_{\mu\nu}= (\frac 12 g_{\mu\nu}\nabla^2- \nabla_\mu\...
- 56.5k
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4
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How do gravitational fields combine together in GR?
When we have 2 massive bodies coming close together say 2 black holes or 2 massive stars, how do their respective metrics/spacetime curvature combine in the space in between them?
Do we write
$$G_{\mu\...
- 2,042
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1
answer
75
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Green's function in General Relativity: when can we express the Ricci tensor as an operator?
One can express the Riemann tensor acting on a vector field $V^\mu$ as
$$R^\sigma_{\mu\rho\nu} V^\mu = \left(\nabla_\rho\nabla_\nu - \nabla_\nu\nabla_\rho\right)V^\sigma,$$
so we must be able to ...
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Derivation of metric flatness locally
In Einstein Gravity by Zee chapter I.6, he discussed the local flatness of the metric. There are two steps in what he did. First, he showed that the metric at a point can always be written as a flat ...
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2
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73
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Max distance in 2d and 3d positively curved space
On a 2d positively curved space (surface of a sphere), using polar coordinates, we have $ds^2 = dr^2 + R^2 sin^2(r/R) d\theta^2$. I understand we can calculate circumference by going around $\theta$ ...
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In a Spatially One-Dimensional Universe, is a Minkowski Space-Time Diagram accurately graphable if we include the effects of "gravity"?
I've been working on studying Special Relativity and General Relativity for the past few years. As I think we all know, GR gets a lot more complicated than SR, and my knowledge is limited. I am very ...
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58
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Why cant I measure proper length on spacetime curvature with the following formula? [closed]
I'm struggling right now from the definition of proper length along spacetime curvature, it is said as I found online the length that object covered on his spacetime rest frame , so why cant I use the ...
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Calculating Ricci tensor given the metric - undergraduate GR [closed]
I’ve been trying to solve the following problem:
Consider an $(N + n + 1)$-dimensional spacetime with coordinates $\{t, x^I, y^i\}$, where $I$ goes from 1 to $N$ and $i$ goes from 1 to $n$. Let the ...
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68
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Extrinsic Curvature Calculation on the Sphere
Given the following 2+1 dimensional metric:
$$ds^{2}=2k\left(dr^{2}+\left(1-\frac{2\sin\left(\chi\right)\sin\left(\chi-\psi\right)}{\Delta}\right)d\theta^{2}\right)-\frac{2\cos\left(\chi\right)\cos\...
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2
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131
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Homogeneous and Isotropic But not Maximally Symmetric Space
Is this statement correct: "In a homogeneous and Isotropic space the sectional curvature is constant, while in a maximally symmetric space the Riemann Curvature Tensor is covariantly constant in ...
- 1,248
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3
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679
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Constant curvature on a sphere?
$ds^2 = \frac{1}{1- r^2}dr^2 + r^2d\theta^2$ denotes a 2d spherical surface and it should have a constant curvature. The Riemann curvature tensor components are linear in their all 3 inputs. Since the ...
- 1,248
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Alternative definition of the Ricci Scalar
I came across this definition of the Ricci Scalar on its Spanish Wikipedia page:
$$R=-g^{\mu\nu}\left(\Gamma_{\mu\nu}^{\lambda} \Gamma_{\lambda\sigma}^{\sigma} - \Gamma_{\mu\sigma}^{\lambda}\Gamma_{\...
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380
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How to find that there is a conical singularity in the BTZ black hole?
Considering a non-rotating and non-charged 2+1 dimensional black hole, known as the BTZ black hole which obtained by adding a negative cosmological constant $\Lambda=-\frac{1}{l^2},l\ne0$ to the ...
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How to mathematically describe the process of spacetime curvature?
I guess as a result of the energy-momentum tensor $T_{\mu\nu}$ coupling to a flat Minkowski metric, $\eta_{\mu\nu}$, the flat metric can become that of a curved spacetime, $g_{\mu\nu}$. How can one ...
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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 ...
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64
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How does a proper antichronous transformation change the geodesic equation?
I want to apply a proper antichronous transformation to the geodesic equation in General Relativity and check if it is even or odd.
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1
answer
64
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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). ...
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781
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How is the Ricci scalar the trace of the Ricci tensor?
The Ricci scalar is the uncontracted version of the Ricci tensor $R=R^{\mu}_{\mu}=g^{\mu\nu}R_{\mu\nu}$. Carrol describes the Ricci scalar as being the trace of the Ricci tensor, but I do not ...
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2
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189
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Can $\mathbb{R}^4$ be globally equipped with a non-trivial non-singular Ricci-flat metric?
I'm self-studying general relativity. I just learned the Schwarzschild metric, which is defined on $\mathbb{R}\times (E^3-O)$. So I got a natural question: does there exist a nontrivial solution (...
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Can there be black holes in a 1+1D spacetime? [duplicate]
In 2D Einstein tensor is always zero, that means no mass (or cosmological constant or other stress energy tensor component) is allowed. Nevertheless, we can get nonzero Riemann, even nonzero Ricci ...
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How to solve nonlinear diff eq for a general Schwarzschild metric?
So I have a general form of a spherically symmetric metric:
$$ds^2 = -g(r)_t \, dt^2 + g(r)_r \, dr^2 + g(r)_s (d\theta^2 + \sin^2 \theta \, d\Phi^2)$$
$$ R_{\theta\theta} = \frac{-g'_s g'_t g_r + ...
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2
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536
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Zero Einstein Tensor in 4D
In 2D the Einstein tensor is always zero, and we can easily get solution with non-zero Ricci tensor but zero Einstein tensor. But is it possible in 4D? Can we get a space-time with zero Einstein ...
- 1,248
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56
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Is the Bianchi-identity conformally invariant?
I am trying to show that for a conformal transformation $\tilde{g}_{ab} = \Omega^2 g_{ab}$ the divergence of the non-physical Einstein-tensor $\tilde{G}_{ab}$ (i.e. the Einstein tensor corresponding ...
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1
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193
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Disentagling coordinates and curvature?
While trying to understand General Relativity, I'm struggeling with disentangling coordinates and curvature.
The metric tensor contains information on both: coordinates as well as curvature.
Curvature ...
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125
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How do you show that Einstein's tensor simplifies to the unitary form in geometric algebra?
In David Hestenes' Gauge Theory Gravity paper (RG), he claims that
$$G^\beta = \tfrac{1}{2}(g^\beta \wedge g^\mu \wedge g^\nu) \cdot R(g_\mu \wedge g_\nu)$$ in geometric algebra (specifically, ...
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209
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Why isn't the curvature scale in Robertson-Walker metric dynamic?
$$ds^2=-c^2dt^2+a(t)^2 \left[ {dr^2\over1-k{r^2\over R_0^2}}+r^2d\Omega^2 \right]$$
This is the FRW metric, here k=0 for flat space, k=1 for spherical space, k=-1 for hyperbolic space. $R_0$ is the ...
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Is it possible to describe every possible spacetime in Cartesian coordinates? [duplicate]
Curvature of space-time (in General Relativity) is described using the metric tensor. The metric tensor, however, relies on the choice of coordinates, which is totally arbitrary.
See for example ...
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Why does the Weyl tensor not show up in the Einstein field equations?
In the Einstein field equations, the only tensor that shows up is the Ricci tensor and the metric tensor, together with the Ricci scalar. The Weyl tensor though is a tensor that is a part of the ...
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259
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How many independent degrees of freedom does the metric tensor have in vacuum (at every point)?
A field of metric tensors fully characterises the curvature of a vacuum space-time. (For example, the spacetime between some single point masses which are themself not part of the manifold)
The metric ...
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Can we use vielbeins on curved space?
My question is: Can we use vielbeins on (let's say ) an anti de-sitter space?
This is why I am confused:
To couple fermions with gravity in curved space, using vielbeins is a well-known approach. So ...
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261
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Deriving the Ricci tensor on the flat FLRW metric
I am currently with a difficulty in deriving the space-space components of the Ricci tensor in the flat FLRW metric
$$ds^2 = -c^2dt^2 + a^2(t)[dx^2 + dy^2 + dz^2],$$ to find:
$$R_{ij} = \delta_{ij}[2\...
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1
answer
150
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How do you relate $\Omega_{k}$, the curvature term in the FLRW metric, to the radius of curvature?
I have assumed, for reasons a bit too detailed to go into here, that if $\Omega_{k}$, the curvature term in the FLRW metric, is equal to 1, then the radius of curvature is equal to 13.8 billion light ...
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1
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Indices of $(\text{Riem})^3$?
This question relates to writing higher curvature terms in momentum space with respect to GR as an effective field theory.
I know that $R_{\alpha\beta\mu\nu} \sim \partial_\beta\partial_\mu h_{\alpha\...
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2
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241
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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 ...
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2
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Meaning of zeros in the metric tensor
I'm trying to find the $g_{0i}$ components of the metric I mentioned here, but it has turned extremely difficult. My current strategy is to equate Ricci tensor components gotten from the Christoffel ...
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4
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381
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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?...
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3
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What is Dirac's reasoning when showing the curvature vanishing implies we can choose rectilinear coordinates?
In section 12 of Dirac's book "General Theory of Relativity" he is justifying the name of the curvature tensor, which he has just defined as the difference between taking the covariant ...
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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 $R_{abcd}$?
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1
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Simplifying the contraction of the riemannian curvature tensor with a timelike projection matrix
I'm reading this short piece "Physical and Geometric Interpretations of the Riemann Tensor, Ricci Tensor,and Scalar Curvature" (pdf), and I'm having trouble grasping one the simplifications ...
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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 ...
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89
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Under what conditions would Ricci tensor have the trace values $(+---)$ and rest 0's?
Under what conditions would Ricci tensor have the following values:
$R_{\mu\nu} = 0$ where $\mu \not= \nu$
$R_{00} = 1$
$R_{\mu\nu} = -1$ where $\mu = \nu$ (and non-zero)
- 387
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1
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258
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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 ...
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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?
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1
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103
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1+1D simple vacuum EFE solution
Can there be any solutions for simple vacuum Einstein Field Equations in 1+1D (1 space and 1 time dimension) i.e $R_{\mu\nu} = 0$ except for flat space?
I tried different combinations of random ...
- 1,248
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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
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191
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Combining two spacetimes of different curvature
This is based on a question by someone else on this site:
Combining metric tensors/curvature tensors
They asked:
Consider a particle which causes a metric $g_{\mu\nu}$
on an otherwise Minkowski ...
- 613
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84
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Jump in Extrinsic Curvature
In the thin shell formalism, the components of the surface stress-energy tensor are found using the next equation:
\begin{equation}
S^{a}_{\, \, \, b} = const\left([K^{a}_{\, \, \, b}] - \delta^{a}_{\,...
- 1,327
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1
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140
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What's the physical curvature scale $R_0$ in the FLRW metric?
I'm studing the FLRW metric using Daniel Baumann's book, Cosmology (2022), in this book the author derived the FRLW metric using the following equations:
$$ dl^2 = \textbf{dx}^2 \pm du^2 $$ and $$ \...
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1
answer
525
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How do I use the Schwarzschild metric to calculate space curvature and time curvature seperately?
I want to understand the math behind the idea that around Earth time dilation accounts for 99.99% of gravity, while around a black hole it only accounts for 50% of gravity while space curvature ...
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