Questions tagged [linearized-theory]
The linearized-theory tag has no usage guidance.
159
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Can linear frame dragging cause gravitational dipole radiation?
I have just learned that linear frame dragging exists in General Relativity. I have also seen simulations where a periodically accelerated and decelerated mass causes a sort of gravitational dipole ...
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92
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Liénard-Wiechert potentials in General Relativity
In linearized gravity, we have.
$$
\Box \bar{h}_{\mu \nu} = -\frac{16 \pi G}{c^4} T_{\mu \nu}
$$
Solving the equationg for a point mass, with a trajectory $\mathbf r_0(t')$:
$$
\rho(\mathbf r', t') = ...
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What is the regime of validity for linearized gravity?
A common formulation of the approximation is to assume the metric is give by
$$g_{\mu\nu}= \eta_{\mu\nu} + h_{\mu\nu},$$ where $\eta_{\mu\nu}$ is the Minkowski metric and the components of $h_{\mu\nu}$...
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Gauge invariance of linearized gravity with an arbitrary background spacetime
Consider here a background metric $g_{\mu\nu}$, we impose a perturbation $g_{\mu\nu}+\epsilon h_{\mu\nu}$ with $\epsilon\ll1$. Then we can write down the modified Einstein-Hilbert action with zero ...
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Vibrational Modes and Imaginary Frequencies of a Three Spring System
This question is an extension of the one I posted a few days ago: Rigid Body and Two-Spring System and the Lagrangian.
I am attempting to find the vibrational modes and their frequencies of an ...
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2
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0
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41
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Orbital phase and gravitational wave phase
I am trying to understand the relation between the orbital phase of binary and the phase of the gravitational wave when expressed as spin-weighted spherical harmonics.
The metric perturbation can be ...
- 1,064
2
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Linearized gravity in BTZ black hole
I'm currently studying this article on traversable wormholes in an eternal BTZ black hole. In the first pages the authors say that is instructive to check that a small spherically symmetric ...
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2
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135
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Linearization of an expression
I'm doing a physics lab and I have a question that asks me to linearize the expression of time (it gives me the equation for time) in function of the mass.
I don't want any solution, just want to know ...
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0
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Why does the dielectric current density operator is twice its value?
In second quantization, the time-independent Hamiltonian for free fermions is written as (forgetting its effect on the spin)
$$\mathcal{H}_0=\sum_\sigma\int\mathrm d^3 \mathbf r\; \Psi^\dagger_\sigma(\...
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Does Hooke's Law apply to all springs?
I understand that Hooke's Law is $F=-kx$, and that this law only applies when a spring is not "overstreched." However, does Hooke's Law apply to all springs, or only simple harmonic ...
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227
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Second-order Lagrangian of Einstein-Hilbert action
I'm having trouble deriving the equation (44) of https://arxiv.org/abs/1710.08863 .
The question is how to get the second-order lagrangian of the Einstein-Hilbert action, i.e.
\begin{equation}
\...
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3
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256
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How to justify this small angle approximation $\dot{\theta}^2=0$?
Suppose the equation of motion for some oscillating system takes the following form:
$$\ddot{\theta}+\dot{\theta}^2\sin\theta+k^2\theta\cos\theta=0$$
Applying small angle approximation to $\theta$ ...
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1
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Should the linearized field equations of GR with cosmological constant be gauge-invariant?
Say I have a solution to Einstein's field equations (EFE) with cosmological constant (CC)
$$
G_{a b}[g] + \Lambda g_{a b}=T_{ab}[g,\Phi]
$$
and want to find a perturbative solution $g_{a b} + \delta ...
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1
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78
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Is the massive Fierz-Pauli action invariant under changes of coordinates?
The action for linearlized gravity on a curved background has the form
$S_{LinGrav}=\frac{1}{16 \pi G} \int d^{4} x \;\sqrt{-g}\;\left[-\frac{1}{4} \nabla_{\rho} h_{\mu \nu} \nabla^{\rho} h^{\mu \nu} +...
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General relativity 2 particle problem with negative mass
I am looking at a problem of 2-particle system of which one has negative mass. I have situation described on Wiki under section "Runaway motion". Prticulary, if we assume, that negative mass ...
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D'Alembertian operator in linearized gravity
In linearized gravity, where we take a general background metric $g$ with perturbation $h$, the linearized Einstein equations become
$$-\square h_{\alpha\beta}+\nabla^{\delta}\nabla_{\alpha}h_{\beta\...
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What does it mean to linearize the trace of Gauss-Codazzi relations?
What does it mean to linearize the trace of the Gauss-Codazzi relations?
The equations referenced above are the following:
Reference to the paper: https://arxiv.org/abs/hep-th/0511096
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Is a perturbation of a tensor field a tensor field?
Let say I take some $2$-tensor field $T_{\mu\nu}$ on some pseudo-Riemannian manifold. Now, often, we are interested in its linearization, which means that we take a family of tensor fields $T_{\mu\nu}(...
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Derive Linearized Einstein's equation from Lagrangian approach
Given the Hilbert action:
$$
S_{H}=\int \sqrt{|g|}R d^{n}x
$$
and the metric written in terms of Minkowski and perturbed metric:
$$
g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}.
$$
I am able to derived the ...
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95
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Problem with linearization of Einstein-Hilbert action in de Sitter background
For some purpose, I have to calculate the well-known linearization of the Einstein-Hilbert action in the de Sitter background. I encountered a problem: assuming the de Sitter metric, my resulting ...
- 2,332
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1
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228
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Einstein-Hilbert Lagrangian in linearized gravity
The Einstein-Hilbert Lagrangian is:
$$\mathcal{L}_{EH}=\sqrt{-g} R$$
where $g={\rm Det}[g_{\mu\nu}]$ and $R$ is the Ricci scalar. In linearized gravity $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and
$$\...
user255856
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1
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Gravitoelectromagnetism: How far does the analogy go?
In weak gravitaional fields, we can write equations analogous to the Maxwell equations: Gravitoelectromagnetism.
Do the gravitoelectric field and the gravitomagnetic field transform like components ...
- 2,053
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1
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764
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"Lorentz gauge" or "Lorenz gauge"?
In electrodynamics there is a gauge condition named after Ludvig Lorenz: $$\partial^\mu A_\mu = 0.$$
In general relativity we also have a gauge condition defined as follow:
$$\partial_\mu \gamma^{\mu\...
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votes
3
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What do we mean when we say gravitational waves are non-linear and do not superpose like EM waves?
I have read this question:
Now it's not actually true that general relativity obeys a law of superposition, but it is an extremely good approximation for a small-amplitude gravitational wave passing ...
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Why does the graviton polarization satisfy $\epsilon_{ij}(\mathbf{k},\lambda)\epsilon^{ij}(\mathbf{k},\lambda') = 2 \delta_{\lambda\lambda'}$?
I am reading the paper ``Graviton Mode Function in Inflationary Cosmology'' by Ng (link here). The graviton $h_{ij}$ is here expanded (in the TT gauge) where
$$
h_{ij}(x) \sim \epsilon_{ij}(\mathbf{k},...
- 3,463
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3
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Issue expanding $\sin \theta$ about $\theta_{eq}$
Quoting a textbook:
$$(m_1 + 2m_2\sin^2\theta)\ddot\theta = m_1\Omega^2\sin\theta\cos\theta - \frac g L (m_1 + m_2)\sin\theta.\tag{10}$$
We can simplify this expression a bit by relating $\frac g L (...
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Harmonic Gauge in linearized GR and meaning of coordinate system
In Linearized gravity one can perform coordinate transformations
$$x^\mu \rightarrow x'^\mu=x^\mu+\xi^\mu(x)~~~~~~\text{with the condition } \Biggl|\frac{\partial \xi^\mu(x)}{\partial x^\nu}\Biggr|\...
- 2,505
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An equation satisfied by harmonic coordinates in a Schwarzschild spacetime
(Full disclosure, this is very much a "fish out of water" type situation: I'm working on something in an area very far away from the stuff I usually work with, so this is hopefully something ...
2
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2
answers
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Fierz-Pauli action as an effective action from Einstein-Hilbert acition?
The Fierz-Pauli action
$$
S=\frac{1}{16 \pi G} \int d^{4} x\left[-\frac{1}{4} (\partial_{\rho} h_{\mu \nu})( \partial^{\rho} h^{\mu \nu}) + \frac{1}{2} (\partial_{\rho} h_{\mu \nu}) (\partial^{\nu} h^{...
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2
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137
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Linearized theory and gravitational waves
I've been reading the chapter about gravitational radiation of Schutz's book.
In one of the sections, he begins with the linearized Einstein's equations and tries to find an intuitive solution:
$$(-\...
- 21
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0
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How is the gravitoelectromagnetism approximation of GR valid if it seems to yield unstable solutions?
In the gravitoelectromagnetism approximation of GR, we have equations analogous to Maxwell's equations with some sign changes. As pointed out in another post of mine, this leads to unstable run-away ...
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Is it OK to do this manipulation at the partition function level? (auxiliary fields in quadratic gravity)
Background
I am working with the following action in the Euclidean signature ($C^2$ is the Weyl quadratic term):
\begin{equation}
S_B = -\frac{1}{2\kappa^2}\int d^4x\sqrt{g}\left(2\Lambda_C+\zeta R-\...
- 2,332
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votes
3
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views
What are the differential equations that model a self-propagating gravitational wave in space-time?
Light is a self-propagating wave, but it's very complicated.
Imagine, if you will, a wave in space-time that by assumption was self-propagating like light, except that it was a gravitational wave.
...
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2
answers
483
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Understanding the derivation of the Newtonian limit of GR
I'm having difficulty understanding some of this derivation. The relevant information, as I understand it is:
We've let the lorentzian metric be a perturbation of the minkowski metric:
$$g_{\mu\nu} = \...
- 47
3
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1
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Deriving Gravitoelectromagnetism Lorentz Force Equation
I'm attempting a problem from Wald (Chapter 4, problem 3a) and having quite a bit of trouble.
Problem
The text states, "If we assume that the time derivatives of $\bar{\gamma}_{ab}$ are ...
0
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1
answer
75
views
Why Fourier transform for linearized NS?
I'm reading a paper and has the linearized NS equation and follows it by getting the solution through a Fourier transform. What is the thought behind this? Meaning, why use a Fourier transform?
$ \rho\...
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1
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Details of using flat metric to raise/lower indices in linearized GR. I'm getting first order discrepancies
This question is about the use of the unperturbed (Minkowski) metric $\eta_{\mu\nu}$ (and its inverse $\eta^{\mu\nu}$) to raise and lower indices in linearized gravity. There are already several ...
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Issues carrying out Taylor Expansion for small oscillations [closed]
I have an equation of motion for a pendulum: $\ddot{\theta}+\frac{g}{l}\sin(\theta)-\dot{\phi}^2\sin(\theta)\cos(\theta)=0$
I want to taylor expand this equation for small oscillations, i.e.: $\theta =...
11
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1
answer
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views
Why do people care so much about 'linear response theory'?
In the lectures I've heard linear response theory was introduced multiple times, however I can't tell how this is any different from solving an inhomogeneous PDE with a greens function and there's ...
- 1,943
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Why do we ignore the second-order terms in the following expansion?
Consider the expansion done for the kinetic energy of a system executing small oscillations as done in Goldstein:
A similar series expansion can be obtained for the kinetic energy. Since the ...
1
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1
answer
144
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Perturbation theory covariant derivative
Let $\varphi$ be a scalar field and consider the metric $g_{\mu \nu}=\eta_{\mu \nu} + h_{\mu \nu}$. I want to compute $\nabla_\mu \varphi \nabla^\mu \varphi$ to first order.
$$\nabla_\mu \varphi \...
- 1,157
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Linearization of 1D maps about a fixed unstable point [closed]
Recently, I was going through the paper Controlling Chemical Chaos in a three variable autocatalator system, by Peng et al. Here are the references
Although I have been introduced to 1D maps and the ...
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2
answers
121
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Raising/lowering indices in linearized GR
In linearized general relativity, we have the unperturbed metric and the perturbed metric. In all textbook treatments, they say that they are going to raise and lower indices with the unperturbed ...
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Einstein-Hilbert Action for Linearized Gravity
Following a derivation for the action in linearized gravity I have reached the point where I have the following expression
$$S = \frac{1}{16 \pi G} \int d^{4}x \sqrt{-g} R.$$
I know that
$$R = \...
- 21
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vote
1
answer
401
views
Riemann tensor in linearized theory of gravity [closed]
Here I present some approximations to Christoffel, Riemann, Ricci tensors when the following perturbed metric is taken into account $g_{\mu \nu} \approx \eta_{\mu \nu} + h_{\mu \nu} + \mathcal{O}(h_{\...
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Projection on transverse traceless Gauge
I am reading this book about Gravitational waves.
On page 10 and 11 Maggiore says that
Given a plane wave solution $h_{\mu\nu}$ propagating in the direction $\hat{n}$, outside the sources, already in ...
user255856
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Prove of conservation of energy and momentum in linearized Einstein Field Equation
From the Linearized Einstein Field Equation, we have $\Box\bar{h}_{\mu \nu} =-16\pi GT_{\mu \nu}$.
How can I obtain conservation of energy and momentum, $T_{\mu \nu},^{\nu}=0$, from the previous ...
- 11
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1
answer
322
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Hilbert gauge and gravitationl waves
I am studying the gravitational waves as a conseguence of the linearized (vacuum) field equation of general relativity.
I have a problem when we write the linearized field equation in the form $\Box ...
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0
answers
97
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Consistency condition in the context of gravitational waves
Reference: archive.org/details/GeneralRelativity/page/n250/mode/1up
I am studying the phenomenon of gravitational waves as one of the conseguences of the linearized Einstein field equation.
In order ...
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Transverse-traceless gauge: Why the traceless condition?
I'm right now following a course on GR and I arrived to the gravitational waves part. Letting the metric be that of the plane Minkowski space with a small perturbation:
$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\...
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