Questions tagged [linearized-theory]

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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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1answer
50 views

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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1answer
79 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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2answers
60 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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1answer
36 views

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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1answer
38 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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1answer
108 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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1answer
37 views

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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1answer
108 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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1answer
71 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 ...
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1answer
54 views

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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1answer
64 views

What is the reason gravitational waves propagate at speed of light? [duplicate]

I know Maxwell equation shows that it is the electric and magnetic properties of empty space which is why speed of light is constant, but what about gravitational waves? I have visited many answers ...
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3answers
219 views

Linearized gravity and local Lorentz symmetry

Action for linearized gravity is well-known, see for example David Tong: Lectures on General Relativity: $\mathbf{The\;Fierz-Pauli\;Action}$ The linearised equations of motion can be derived ...
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3answers
66 views

I can't understand one of deduction in Simple Harnomic Motion, can anyone help?

source:http://farside.ph.utexas.edu/teaching/336k/lectures/node18.html#e4.8 in order $x=0$ to be a stable equilibrium point we require both $$f(0)=0$$ and $$\frac {df(0)}{dx}<0$$ Now, our particle ...
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1answer
71 views

Weak Field Limit of Einstein's Equation

I am learning about harmonic gauge for weak field limit of Einstein's equation and have some problems with the tensor calculus invovlved. Consider the weak field approximation $$g_{\mu\nu}(x)=\eta_{...
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1answer
140 views

Stuck Solving MTW Gravitation Problem 20.5

I am stuck on exercise 20.5 part a) from Misner, Thorne, and Wheeler's Gravitation chapter 20. The Einstein summation convention is used throughout this post. Problem Statement Calculate $t^{\...
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44 views

Why linear response theory is exact for quadratic Hamiltonian?

In linear response theory, we consider the Hamiltonian $H(t)=H_0 + \theta(t)H'(t)$, where $H'$ is a perturbation that is turned on $t=0$. A standard result is that for an observable $A$, $$ \langle A(...
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72 views

Gravitons from linearized gravity

Linearizing gravity as follows: $$ g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu} $$ Up to this point, everything is a 4x4 matrice. How does one eventually recover a spin-2 particle which, according to https:/...
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2answers
125 views

Contracting Riemann Tensor Troubles

It has been several years since I looked at General relativity, and I am trying to brush up on it because it was always interesting and I am in need of it for my research. Specifically, I am looking ...
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1answer
53 views

Linearizing the Lugiato-Lefever Partial Differential Equation

Problem Statement: Given the Lugiato-Lefever equation, linearize the equation and determine the dynamics near a stationary solution by looking for a stationary solution with a small perturbation. ...
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54 views

General Relativity Lorentz-like equation

In the literature, it says that, in the weak-field, $$g_{µν} = η_{µν} +h_{µν},$$ slow-motion limit, the Geodesic equation reduces to the Lorentz-like equation. Can anyone explain this?
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1answer
383 views

Perturbed Ricci tensor due to metric perturbation i.e. $R^{(2)}_{\mu\nu}[h]$ in Linearized theory of Einstein field equation

This is an equation (7.153) from Chapter-7 of Sean Carroll's An introduction to General Relativity: Spacetime and Geometry book. I think all of you who studied GR and went thorugh Carroll's book have ...
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1answer
146 views

Reference on quantization of linearized gravity

I often see claims saying that quantization of linearized gravity can be done. Is there some standard references for it? such as original papers/ review papers / textbooks etc.
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Perturbation to the flat space metric

from the geodesic equation for non-relativistic case where $$v_i\ll c$$ $$\frac{dx^i}{dt}\ll1,{\rm for }\ c =1$$ $$\frac{dx^i}{d\tau}\ll\frac{dt}{d\tau}$$using this the geodesic equation for proper ...
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3answers
250 views

How to get space component of weak field (linearized) metric?

For Minkowski space with a weak gravitational field the metric takes the form $$ ds^2 = (1+2\phi/c^2)c^2dt^2 -(1-2\phi/c^2)(dx^2+dy^2+dz^2), $$ where $\phi$ is the Newtonian gravitational potential. ...
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3answers
181 views

Is Stokes equation a reduction of Navier-Stokes equations?

The following Stokes problem: $$\begin{cases}-\nu\Delta u+\nabla p=f&,\textrm{in }\Omega\\ \nabla\cdot u=0&, \textrm{in } \Omega\end{cases}$$ is a reduction of the Navier--Stokes equations? ...
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1answer
126 views

Metric tensor in linearized gravity [duplicate]

I'm studying linearized gravity, for which the weakness of the gravitational field is expressed as ability to decompose the metric into the (flat) Minkowski metric plus a small perturbation, i.e. $$g_{...
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1answer
195 views

Constraints in general relativity

In this review on inflation, on Pg. 135, Baumann talks about the energy and the momentum constraints for gravity. Are these equations the $G_{00} = T_{00}$ and $G_{0i} = T_{0i}$ components of the ...
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2answers
234 views

How do you derive a quantum field theory from a spacetime metric?

What are the first steps in converting a metric into a quantum field theory? I know roughly what to do once I have a pair of non-commuting operators, but how do I get to that point? Specifically, I'd ...
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1answer
752 views

Linearizing the Einstein-Hilbert action; shortcuts?

I am interested in linearizing actions that are constructed out of geometrical objects. By this I mean perturbing the metric (or vielbein) and keeping in the action terms which are quadratic in the ...
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1answer
43 views

What is a quasistationary approximation

I was reading an article which states : The linear-stability analysis for this system can be performed in complete generality; but it will be best for purposes of this review to go directly to ...
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1answer
4k views

How can I Derive the Equation for Coefficient of Linear Thermal Expansion?

I know the relationship between change in temperature and change in length. When the ambient temperature around any substance is increased, its length increases. This is due to molecules gaining more ...
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1answer
131 views

Gravitational waves energy source in linearized theory

By linearizing the metric in the following way (approach in most textbooks): $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\text{ with } |h_{\mu\nu}|\ll 1$ and choosing the transverse-traceless gauge a wave ...
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1answer
140 views

Can anyone explain the harmonic oscillator (in context to quantum mechanics) 2.3 (Griffiths) using Taylor series?

At the end he concludes $V(x) = V''(x_0)(x-x_0)^2$. How does he get to know that the rest are $0$? How does he conclude $V''(x_0) = k$. Please try to explain in easy ways and tough vocabulary. I don't ...
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2answers
218 views

meaning of “sufficiently small” in approximations to behaviour of a Harmonic Oscillator

So in my classical mechanics book it states: "For any sufficiently small displacement, any system of this kind behaves like a harmonic oscillator." When discussing SHO. So I am curious what is ...
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1answer
455 views

Why do materials obey Hooke's law? [duplicate]

Why do materials extend proportionally to the force exerted on them (Hooke's law)? I thought that when materials are compressed or extended under force, their atoms become closer or further apart; ...
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1answer
239 views

Factor of 4 (or 2) in the gravitoelectromagnetic (GEM) Lorentz-force law. Which is correct? Why is it there?

I realize that the Gravitoelectromagnetic equations (GEM) are derived from the Einstein field equation (EFE) in the degenerate case of reasonably flat spacetime, which is the case for the propagation ...
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1answer
274 views

How can I find the metric in weak field limit for specific theory?

What is the general approach to finding a modified version of Poisson equation by means of the weak field limit of a specific gravitational theory? What is the first step? Can you introduce the main ...
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1answer
131 views

In what physical situations is the weak-field limit invalid?

in the weak-field limit gravitation is described by a symmetric tensor field $h_{μν}(x)$ in flat spacetime. Linear theory suffices for nearly all experimental applications of general relativity ...
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66 views

Has anyone ever tried to derive gravitoelectromagnetic waves equation?

Has anyone ever tried to derive gravitoelectromagnetic waves equation? As we know, there is Maxwell-like equation in gravity. Has anyone here ever formulated gravito electromagnetic waves equation ...
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0answers
203 views

Effective stress-energy tensor for a gravitational wave, compared to static semi-Newtonian case

There is a calculation that I had been thinking for a long time of working out to my own satisfaction, both because of its intrinsic importance and because it seemed like it would be fun. This was to ...
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1answer
750 views

What are the Maxwell's equations for gravitational waves?

Maxwell's four equations can be used to describe the propagation of electromagnetic waves. What is the equivalent for gravitational waves - if that question makes sense?
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4answers
505 views

Why change in resistivity is proportional to the original resistivity?

When there is a temperature change $\Delta T$, the change of resistivity is (1) proportional to $\Delta T$ (2) proportional to the original resistivity $\rho_0$ Hence we can define the temperature ...
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1answer
333 views

Why is the Ricci scalar non-zero in this case?

The Einstein equations can be written as (1): $$R_{ab}-\frac{1}{2}Rg_{ab} = -8\pi GT_{ab}$$ or by contracting the above equation with the metric tensor and resubstituting: (2) $$R_{ab}=8\pi G(\frac{1}...
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2answers
410 views

Trace-reversed EFE and linearized gravity

I have a question about the linearized Einstein Field Equations, and in particular about the Newtonian limit. It goes as follows. If one uses the trace-reversed form of the EFE for the 00-component ...
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2answers
188 views

How to use Ashtekar's variables in classical gravitational physics?

I have often heard of Ashtekar's variables in General Relativity, because of the naturalness with which they would allow a canonical formulation of gravity, useful for a hypothetical quantum gravity ...
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0answers
501 views

Gauge transformation of trace-reversed metric perturbation

This question is in reference to Exercise 30.4.2 in Thomas Moore's A General Relativity Workbook, which asks you to show that a gauge transformation of the trace-reversed metric perturbation $H_{\mu\...
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91 views

A question about the metric in linearised gravity

In linearised gravity the usual approach is to perturb the metric around some fixed background (often taken to be Minkowksi). My question is, does one literally perturb the metric tensor itself, i.e. $...
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1answer
231 views

Rationale behind the linearised Navier-Stokes equation

Some applications of fluid dynamics consider the linearised Navier-stokes equation where the advection term $(\vec{u}\cdot\vec{\nabla})\vec{u}$ is dropped. I am trying to build a convincing argument ...
3
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0answers
58 views

Linearized variation of $\sqrt{-g}(\nabla^\mu \xi^\nu - \nabla^\nu \xi^\mu)$

The Noether current associated with invariance of the Einstein-Hilbert action, $$ S=\int d^dx \sqrt{-g}\,R[g_{\mu\nu}]\,, $$ under infinitesimal diffeomorphisms $\delta_\xi g_{\mu\nu}=\nabla_\mu \xi_\...