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The "small amplitude" assumption in the derivation of the wave equation for the string

I am reading about the wave equation for transverse waves in a string from the book Mathematics of wave propagation (2000) by J. Davis. On page 10, just before the derivation of the (one-dimensional) ...
DinoS's user avatar
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0 answers
31 views

Problem with linearized gravity on flat background in spherical coordinates

I am solving the linearized Einstein equations with a flat background in spherical coordinates, i.e $ds^2 = -dt^2 + dr^2 + r^2 d\Omega^2$ and writing $h_{\mu\nu}$ in terms of spherical harmonics. ...
4 votes
0 answers
93 views

Lorentz force error in the present 2024 version of the gravitoelectromagnetism Wikipedia page? [closed]

I noticed that the Gravitoelectromagnetism (GEM) Wikipedia page has been edited recently. The factor of 4 in the GEM Lorentz force equation is now missing. But the GEM field equations are identical to ...
rdryne's user avatar
  • 41
1 vote
1 answer
63 views

Quantised Newtonian potential as an operator in non-relativistic QM [closed]

Suppose we have two slowly moving (effectively static) masses $m_1,m_2$, interacting through gravity, that are not occupying a definite state of position i.e. that matter is being treated quantum ...
Theoreticalhelp's user avatar
0 votes
0 answers
31 views

Linearization of long wave radiation equation

I am basing my reduced order building model on the equations proposed in the article “A methodology for generating reduced-order models for large-scale buildings using the Krylov subspace method” ...
Martin B's user avatar
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0 answers
32 views

On the choice of observables in linear response theory

For linear response theory I need two observables. The idea is to see how the change in one observable changes the other under weak perturbations in equilibrium state. Suppose I want to see the change ...
Rafi Ullah's user avatar
1 vote
1 answer
73 views

Gravitational waves from metric perturbation

I have just been introduced to gravitational waves from metric perturbations and I have some questions about gauge symmetry and solutions in a given gauge. Consider a metric on the form $g_{\mu\nu} = \...
ICOR's user avatar
  • 69
11 votes
5 answers
1k views

How do gravitational waves carry energy when gravitational energy cannot be localised?

I have a very naive question, actually someone asked it and I can't answer. It simply asks that if gravitational energy cannot be localised (we cannot write a pure gravitational energy momentum tensor)...
damaihati's user avatar
  • 415
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1 answer
37 views

Is there a class of phenomena the description of which is not limited to the study of the properties of individual harmonic waves?

There are lots of examples of oscillatory phenomena in nature the description of which boils down to simple harmonic behavior, i.e. to Cosine/Sine/Complex Exp. This answer explains that we use sines ...
shamil khal's user avatar
2 votes
4 answers
181 views

Why the products of $\Gamma$'s of the Ricci tensor can be neglected in linearized GR?

In the linearization of GR, when $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, and $|h| \ll 1$, it is said (for example here) that 'to linear order the “$\Gamma\Gamma$” terms go away' in the formula for ...
Claudio Saspinski's user avatar
14 votes
3 answers
2k views

How do physicists mathematically define gravitational waves?

When one first encounters gravitational waves in a standard GR lecture or a standard textbook like Carroll's "Spacetime and Geometry", they are often "defined" as follows: The ...
Moguntius's user avatar
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0 answers
66 views

How to find the stability of time dependent Lyapunov equation?

After linearization of the nonlinear equations, I want to find the covariance matrix $v$ through the numerical solution of time dependent Lyapunov equation, $$dv/dt=a*v + v*a'+ d,$$ where $a$ is my ...
Ani's user avatar
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0 answers
23 views

Is small perturbation in axial direction directly analogous to radial direction for cylindrical coordinate?

In cylindrical coordinate, the stability for a cylindrical liquid column/ligament can be analysed using perturbation theory by applying small perturbation in radial direction as follow; $$\rho(z,t)=\...
jamill1283's user avatar
0 votes
1 answer
77 views

Are there any "linear" lagrangian systems of interest for which an analytic solution is not obvious?

Out of curiosity, I am interested in Lagrangian dynamical systems that can be expressed in a "linear" manner. By this, I mean that their Lagrangian can be expressed, quadratically, as $$L = \...
Meclassic's user avatar
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4 votes
0 answers
213 views

Justifying the transverse-traceless gauge

For weak gravitational fields, we can assume the metric is some perturbation of flat space: $g_{ab} = \eta_{ab} + h_{ab}$. Following Schutz's argument, you can incorporate a small coordinate ...
Kiwi breeder's user avatar
0 votes
2 answers
251 views

Which of the following expressions for Christoffel symbols is correct?

As shown in the figure, after adding a small perturbation term $h$ to the metric $g$, the expression of Christoffel symbols in an article is Eq. (10) But I think the result should be in the following ...
math's user avatar
  • 13
1 vote
1 answer
75 views

Linearzing technique for the Vlasov equation

I am currently reading a paper on a quantum algorithm to solve the linear Vlasov equation with collisions (https://doi.org/10.1103/PhysRevA.107.062412, A. Ameri, E. Ye, P. Cappellaro, H. Krovi, N. ...
2 votes
0 answers
53 views

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 ...
ramiel46692's user avatar
1 vote
0 answers
109 views

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') = ...
Álvaro Rodrigo's user avatar
1 vote
0 answers
72 views

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 $h_{\mu\nu}$...
yaron kedem's user avatar
1 vote
1 answer
196 views

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 ...
Lain's user avatar
  • 347
1 vote
1 answer
121 views

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 ...
Alex Vaughan's user avatar
2 votes
0 answers
48 views

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 ...
Khushal's user avatar
  • 1,124
3 votes
0 answers
91 views

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 ...
SouthernLion's user avatar
1 vote
2 answers
394 views

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 ...
randomdude's user avatar
5 votes
0 answers
107 views

Why does the dielectric current density operator is twice its value?

In second quantization, the time-independent Hamiltonian for free fermions is written as $$\mathcal{H}_0=\sum_\sigma\int\mathrm d^3 \mathbf r\; \Psi^\dagger_\sigma(\mathbf r) \frac{\hat{P}^2}{2m } \...
Mauricio's user avatar
  • 5,568
12 votes
4 answers
5k views

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 ...
LottaQeustions's user avatar
0 votes
1 answer
443 views

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} \...
gravitone123's user avatar
2 votes
3 answers
495 views

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 $\theta$ ...
Jimmy Yang's user avatar
2 votes
1 answer
141 views

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 ...
JMLF's user avatar
  • 21
3 votes
1 answer
111 views

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} +...
Nik's user avatar
  • 31
7 votes
0 answers
357 views

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". Particulary, if we assume, that negative ...
Vid's user avatar
  • 976
1 vote
2 answers
244 views

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\...
B.Hueber's user avatar
  • 854
0 votes
0 answers
102 views

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
Swahran's user avatar
  • 11
2 votes
1 answer
192 views

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}(...
B.Hueber's user avatar
  • 854
1 vote
1 answer
323 views

Derive Linearized Einstein's equation from Lagrangian approach [closed]

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 ...
wong tom's user avatar
  • 567
0 votes
1 answer
168 views

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 ...
Jeanbaptiste Roux's user avatar
0 votes
1 answer
419 views

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 $$\...
user avatar
2 votes
1 answer
144 views

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 ...
Rd Basha's user avatar
  • 2,141
3 votes
1 answer
2k views

"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\...
JinH's user avatar
  • 126
7 votes
3 answers
4k views

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 ...
Árpád Szendrei's user avatar
2 votes
3 answers
270 views

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},...
QuantumEyedea's user avatar
1 vote
3 answers
243 views

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 (...
realanswers's user avatar
1 vote
1 answer
453 views

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|\...
Sergio's user avatar
  • 2,635
3 votes
0 answers
258 views

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 ...
Rosa Luxemberg's user avatar
2 votes
2 answers
1k views

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^{...
Xiaosheng Yang's user avatar
2 votes
2 answers
180 views

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: $$(-\...
Jiaxiang Zhu's user avatar
2 votes
0 answers
121 views

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 ...
Maximal Ideal's user avatar
3 votes
0 answers
45 views

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-\...
Jeanbaptiste Roux's user avatar
6 votes
3 answers
844 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. ...
StackQuest's user avatar