All Questions
Tagged with perturbation-theory metric-tensor
54 questions
2
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1
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42
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Confusion why Minkowski metric changes sign on metric perturbations [duplicate]
In our textbook, we had
$$g_{µν} = η_{µν} + h_{µν}$$
Then we raised the indices with the Minkowski metric.
$$g^{µν} = η^{µν} - h^{µν}.$$
Why did we get a different sign?
- 89
0
votes
1
answer
58
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Questions For the Expansion of Gravitational Waves
In the appendix B of this paper for effective field theory of gravitational waves, they expand the metric perturbation:
$$g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$$
here deviation $h$ is dimensionless. ...
- 976
1
vote
1
answer
121
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Condition for quadratic correction to first-order perturbation of metric
In Wald's book on General Relativity, the linearized Einstein tensor $G^{(1)}_{ab}$ can be obtained by substituting $g_{ab} = \eta_{ab} + \gamma_{ab}$ in the Einstein equation and ignoring terms that ...
- 330
6
votes
1
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220
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Problem with asymptotic behavior of metric
In the Hamiltonian description of asymptotically flat spacetimes, the metric should deviate at infinity from the flat metric by terms of order 1/r
$$g_{ij} = \delta_{ij} + \frac{\overline{h}_{ij}}{r} +...
- 81
1
vote
0
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42
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Second-order perturbations of gauge field in GR
When expanding a Lagrangian $\mathcal{L}[g_{\mu\nu},A_\mu,\chi]$ to second order in perturbations, the metric is expanded like
$$g_{\mu\nu}\to g_{\mu\nu}+\delta g_{\mu\nu}+\frac{1}{2}\delta g_{\mu}^{\,...
0
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1
answer
68
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Question coming from Cosmological Perturbation
We consider the following scalar perturbation on the FRW metric:
$$ ds^2 = -(1 + 2\phi)dt^2 +2a\partial_i B dx^i dt + a^2 \left( (1 - 2\psi)\delta_{ij} + 2\partial_{ij}E\right) dx^i dx^j $$
where $\...
3
votes
0
answers
136
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Transformation under coordinate transformation of scalar perurbation of FLRW metric
For the past few days I've been studying perturbation in cosmology. More specifically I am now busy with chapter 6 in Dodelson's Modern cosmology. In this book the perturbed FLRW metric which only ...
- 139
0
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1
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95
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Dervation of the first-order Klein-Gordon equation
How to derive the first-order perturbed Klein-Gordon equation:
$$ \square \phi=\left[\frac{1}{\sqrt{-g}}
\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu} \partial_{\nu} \right) \right]\phi=0$$
For a first-...
- 437
2
votes
1
answer
204
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Why is $h_{\mu\nu}$ not a tensor in the perturbed Universe in cosmological perturbation theory?
In the cosmological perturbation theory course per Hannu Kurki-Suonio (2024) :https://www.mv.helsinki.fi/home/hkurkisu/cpt/CosPer.pdf , there is a remark in the text page 5 that puzzles me. The text ...
- 1,255
1
vote
1
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82
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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\...
- 1,073
6
votes
2
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765
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If the metric tensor is unitless, why do its perturbations pick up units of Newton's constant?
If the metric tensor is unitless, why do its perturbation terms pick up units of Newton's constant?
In the following expansion, metric perturbations pick up a factor of $\kappa\propto\sqrt{G}$
\begin{...
2
votes
3
answers
176
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Raising and lowering indices to second order
If I consider a metric perturbed to second order, $$g_{\mu\nu}= \eta_{\mu\nu} + \lambda h_{\mu\nu}^{(1)} + \lambda^2 h_{\mu\nu}^{(2)},\tag{1}$$ how should I raise and lower indices for a generic ...
- 23
-1
votes
1
answer
148
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An excitation of the gravity field
Does the term "excitation of the gravity field" have any meaning ?(or is it just word salad perhaps?)
Other fields seem to have particles associated with them that are described as an ...
- 117
2
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1
answer
227
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Is a perturbation of a tensor field a tensor field?
Let say I take some $2$-tensor field $T_{\mu\nu}$ on some pseudo-Riemannian manifold. Now, often, we are interested in its linearization, which means that we take a family of tensor fields $T_{\mu\nu}(...
- 894
1
vote
1
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408
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Christoffel Symbols for a Perturbed Metric
If a metric $g$ is given by the sum of a background metric $g_B$ and a perturbation $h$ ie. $g_{ij} = g_{Bij} + h_{ij}$, then the difference of the Christoffel symbols for the background metric and ...
- 1,445