Questions tagged [linearized-theory]

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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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3 votes
0 answers
26 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-\...
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6 votes
3 answers
705 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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1 vote
1 answer
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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} = \...
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Weak Field approximation, choosing a diffeomorphism

Consider the EFEs $$G_{\mu\nu}=\frac{8 \pi G}{c^4}T_{\mu\nu}.$$The EFEs are highly coupled non-linear hyperbolic-elliptic second order PDEs that are insanely difficult to solve as they are non linear. ...
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3 votes
1 answer
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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 ...
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0 votes
1 answer
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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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2 votes
1 answer
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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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3 answers
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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 =...
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9 votes
1 answer
656 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 ...
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11 votes
3 answers
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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 ...
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1 answer
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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 \...
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1 vote
1 answer
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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 votes
2 answers
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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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2 votes
0 answers
135 views

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 = \...
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1 answer
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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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2 votes
1 answer
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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 ...
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1 answer
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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 ...
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1 vote
1 answer
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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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3 votes
0 answers
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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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1 answer
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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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1 vote
0 answers
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Linearising two coupled bosonic modes

In Sec. IV of this paper the authors consider the Hamiltonian (Eq. 10) $$ H = \omega_a a^\dagger a + \omega_b b^\dagger b - g_0 a^\dagger a (b + b^\dagger) $$ in the regime $\omega_a / \omega_b \...
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0 votes
1 answer
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Higher-order expansions of a gravitationally coupled system by perturbing the metric

Often in the literature (see e.g. page 142 of https://arxiv.org/abs/gr-qc/9712019), when linearising a gravitational system (pure or matter coupled) around Minkowski space, one writes the metric as $...
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0 answers
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How to measure quantity difference between a nonlinear system of equations and its linearization?

I faced such a problem. I have a nonlinear system for control synthesis and I should compare not only my controllers but also a linear version of my system to describe the legitimacy of this ...
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0 votes
1 answer
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Can we use flat space coordinates in linearized gravity?

I am studying the production of gravitational waves in linearized general relativity. While it is not mentioned anywhere, I am convinced, that the coordinates used in the derivations I have seen, are ...
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0 votes
2 answers
70 views

Software for calculating perturbation effects in general relativity

I need to evaluate the Einstein tensor/ Ricci tensor and others in a perturbed metric. Suppose my metric is $g_{ab} =\eta_{ab} +h_{ab}$, where $\eta$ is the Minkowski metric, and $h$ is a small ...
3 votes
2 answers
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Why can one neglect terms quadratic in derivatives of $h_{\mu\nu}$ in linearised gravity?

In the linear approximation, terms quadratic in the Christoffel symbols are all neglected in the Riemann Tensor. However, these are not quadratic in the $h_{\mu\nu}$ but quadratic in the derivatives ...
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1 answer
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Geometrical linearization in continuum mechanics

In continuum mechanics, we often make use of "physical and geometrical linearization", e. g. during derivation the Navier-Cauchy equations (c. f. https://en.wikipedia.org/wiki/...
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0 votes
1 answer
96 views

1st order approximation to energy-momentum tensor of gravitational field

I was studying linearized gravity and this approximation was given without any derivation. It might be clear for others but I'm quite new on GR and I'm not sure how to get this first order ...
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1 vote
1 answer
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How to find the inverse metric in this case? [duplicate]

Caroll, while proving the Newtonian limit takes $$g_{ab} = \eta_{ab} + h_{ab}$$ He then just writes down the inverse metric to 1st order as $$g^{ab} = \eta^{ab} - h^{ab}$$ I don't see how this ...
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5 votes
1 answer
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Linearized gravity: Derivatives of the metric perturbation

In linearized gravity, the metric is given by the Minkowski metric and a small perturbation, \begin{equation} g^{\mu\nu} = \eta^{\mu\nu}+h^{\mu\nu},\quad |h^{\mu\nu}|\ll 1. \end{equation} Plugging ...
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2 votes
1 answer
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Weak-field Einstein's equation approximation

For Einstein's equation $$ G_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} $$ with $G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R \, g_{\mu \nu}$ where $R_{\mu \nu}$ is the Ricci ...
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1 vote
1 answer
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Physical interpretation of the nil ADM mass of gravitational waves

The ADM mass is defined for any asymptotic flat spacetime. Using cartesian coordinates: \begin{equation}\tag{1} E_{\text{ADM}} = -\: \frac{1}{16 \pi G} \, \lim_{r \, \rightarrow \, \infty}\oint_{\...
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1 vote
1 answer
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How does one add matter coupling terms to the linearized Lagrangian for General Relativity?

In Spacetime and Geometry, Dr. Carroll provides a Lagrangian for Einstein's equations in vacuum assuming that the metric can be written in the form $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. The ...
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0 votes
0 answers
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Linearity of electromagnetism and gravity

If we have a very strong electromagnetic field, it stops being linear, Maxwell's equations stop working ($10^6$ Tesla or $10^9$ Newton/Coulomb); Why can't we say the same thing for gravity? Since the ...
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31 votes
4 answers
3k views

How do photons affect each other gravitationally?

Photons are energy. According to general relativity they should bend space. Assuming two photons pass one another in a large void of empty space how would they gravitationaly affect each other ...
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4 votes
1 answer
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Can I rotate my coordinate system in linearized gravity?

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|\...
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2 votes
2 answers
122 views

Calculating the contravariant form of a perturbed metric

We can write a the covariant form of a perturbed Minkowski background to second order as $$ g_{\mu \nu} = \eta_{\mu \nu} + \kappa h^{(1)}_{\mu \nu} + \kappa^2h^{(2)}_{\mu \nu}$$ where $\kappa$ is ...
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5 votes
1 answer
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Transverse traceless gauge in linearized GR

I'm reading about gravitational waves and I'm wondering how we know we can always go to the transverse and traceless gauge? Going to the de Donder gauge is fine, I can follow that, but then showing ...
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0 votes
0 answers
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Metric Tensor in Weak Field Condition [duplicate]

Suppose that we are in the Weak Field Condition, that is: $$g_{\mu\nu}=\eta _{\mu\nu}+h_{\mu\nu}$$ where $\eta _{\mu\nu}$ is the metric tensor of flat space time (Minkowski spacetime) and $h_{\mu\nu}$ ...
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2 votes
1 answer
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How to derive the hidden symmetry behind linearized gravitation equations?

I am trying to derive the "gauge-like" symmetry of linearized gravitation equation, after deriving the latter heuristically from Newton's universal of gravitation. I am roughly following ...
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5 votes
1 answer
402 views

Raising and lowering indices in linearized gravity

In linearized general relativity indices are raised and lowerd by contracion with the flat space metric tensor $\eta_{\mu \nu}$. I don't really understand why we can do that. In the book gravitational ...
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0 votes
2 answers
109 views

Raising and Lowering Indices of a Perturbed Metric

I have seen in GR that if a metric is a perturbation of some base metric $g^{(B)}_{\mu \nu}$ such that $g_{\mu \nu} = g^{(B)}_{\mu \nu} + h_{\mu \nu},$ then $g^{\mu \nu} = g^{(B) \mu \nu} - h^{\mu \nu}...
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1 answer
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Index gymnastics in weak gravitational field

The metric in a weak gravitational field (TT gauge) is: $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$$ with $$\eta_{\mu\nu}=\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&...
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0 votes
1 answer
177 views

First order expansion of Euler-Lagrange equations

I know that in field theory Euler Lagrange equations are $p_i-d_\mu p^\mu_i=0$. (Classical notations, $p_i=\frac{\partial L}{\partial y^i}, p_i^\mu=\frac{\partial L}{\partial y^i_\mu}$). Being a ...
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0 votes
1 answer
585 views

The action for linearized gravity in a curved background

I'm familiar with the Lagrangian for linearized gravity about a flat background, $$ \mathcal{L} = \frac{1}{2}[(\partial_\mu h^{\mu\nu} \partial_\nu h - \partial_\mu h^{\rho \sigma} \partial_\rho h^\...
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0 votes
1 answer
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Linearised Gravity and Motion of Particles in Background Metric

Let's say we have two point particles as our matter source. Suppose we want to solve Einstein Equation Perturbatively and obtain the gravitational wave at the linear order. Let us expand around ...
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4 votes
1 answer
188 views

Is there a Lorentz invariant approximation to General Relativity?

Since General Relativity is the most accurate description of gravity is there any possible way to derive a Lorentz invariant theory from: $$R_{\mu\nu}-\frac{1}{2} g_{\mu\nu}R+\Lambda g_{\mu\nu}=kT_{\...
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0 votes
1 answer
237 views

Linearized Einstein Field equation in general background

I am trying to compute Linearzed Einstein Field equation in general background. I mean $g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$ and compute R, $R_{\mu\nu}$ and so on. I realized the ...
0 votes
1 answer
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In what way does general relativity predict gravitational waves? [duplicate]

When people say that general relativity predicts gravitational waves, how so?
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