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Questions tagged [linearized-theory]

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1answer
38 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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0answers
43 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
142 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 ...
2
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1answer
71 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
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How to get space component of weak field (linearized) metric?

For Minkowski space with a weak gravitational field the metric takes the form $$ ds^2 = (1+2\phi)dt^2 -(1-2\phi)(dx^2+dy^2+dz^2), $$ where $\phi$ is the Newtonian gravitational potential. You can get ...
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3answers
107 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
70 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
113 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 ...
4
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2answers
153 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 ...
4
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1answer
254 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
37 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
1k 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 ...
4
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1answer
87 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
105 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
131 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 ...
2
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1answer
157 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; ...
2
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1answer
169 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 ...
2
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1answer
173 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 ...
2
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1answer
101 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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0answers
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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
144 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
678 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
293 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 ...
4
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1answer
239 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
262 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
113 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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1answer
78 views

Understanding Boyle's Law and Charles's Law

Boyle's Law is defined as follows: $PV=k$ This implies that $P_{1}V_{1}=P_{2}V_{2}$ is true while temperature and mass of confined gas is constant. This would mean that $P_{2}=P_{1}V_{1}/V_{2}$ ...
3
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0answers
315 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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0answers
68 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. $...
2
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0answers
142 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
46 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_\...
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1answer
72 views

Why don't we write the symmetric term $\eta_{\mu\nu}\partial_\rho\xi^\rho$ in a change of coordinates in linearized gravity?

Reading through this lecture, section (5.3.1): In linearized gravity, when we transform from one set of coordinates to another, the background metric remains Minkowski, but we may think of a metric ...
2
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1answer
61 views

Linearized Euler-Poisson equation

Normalized Euler-Poisson system have the form $$n_t+(nu)_x=0$$ $$u_t+uu_x=\phi_x$$ $$\epsilon^2\phi_xx=n-e^\phi$$ where $u $: ion velocity, $n$: the ion density and $\phi$: electrical potential, The ...
1
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1answer
340 views

Propagating degrees of freedom of gravitational field

I'm following Carroll's Spacetime and Geometry, chapter on Linearized Gravity, pag. 282. He splits up the metric perturbation in scalar, vector and tensor components, writes the Einstein tensor with ...
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1answer
183 views

Linearized gravity: When do we let the metric be $\eta_{\mu \nu} + h_{\mu \nu}$ and when does it reduce to $\eta_{\mu \nu}$?

I am following a standard text on GR. In the chapter on linearized gravity, the metric $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$ reduces to $\eta_{\mu \nu}$ when the metric act on tensor components ...
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1answer
32 views

General relativity, Linealized gravity — which transformations are compatible with the metric?

I am following lecture notes on linearised gravity. The author states that Lorentz transformations are compatible with the metric and begins to Lorentz transform (1) the metric components (2) all ...
2
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1answer
183 views

Tetrad formalism in linearized gravity

I have not seen so far any application of linearized gravity treated using the local orthonormal frame approach. Stuff, such as newtonian limit, gravitational waves, etc. I can totally see the tetrad ...
0
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1answer
44 views

Reference for the integral form of the metric in linearised gravity

Reference wanted: I was looking for a source for the integral form for the weak field metric in terms of the stress-energy tensor, as given in the wikipedia article on Linearised Gravity. The problem ...
0
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1answer
139 views

About linearisation of equations of motion rigid body

I am modeling a system of solid bodies. Consider $\theta \approx 0$ and $\chi \approx 0$. At a certain moment I get the following formula for the angular velocity: $$ \omega = \begin{bmatrix} 0 &-...
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2answers
79 views

Why is stress linearly dependent on strain in elastic materials?

The displacement vector $\mathbf{u}(\mathbf{x}, t) = \mathbf{r}(\mathbf{x}) - \mathbf{x}$ is used to keep track of the motion of the material points in a material. Firstly, we take the gradient of the ...
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1answer
374 views

How use Lorentz Gauge in Einstein tensor?

i have the Einstein tensor in terms h of first order $$ \begin{eqnarray} \label {eq: wf5o} G_{\beta \mu}& = & -\frac{1}{2}\left[ \bar h_{\beta \mu},_{\lambda}^{\;\;,\lambda}+\eta_{\beta\mu}\...
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1answer
93 views

How to find a linear approximation at a critical point for a potential of a conservative field?

I am trying to understand how Arnold derives his tangent equation for branches of a critical point in phase space. Why is the slope at critical point in the phase space- $\sqrt{-U^{''}(\xi)}$ :
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2answers
601 views

Inverse metric in linearised gravity

From what I've read, in the framework of linearised gravity, one perturbs the metric around a Minkowski background, $\eta_{\mu\nu}$, such that $$g_{\mu\nu}(x)=\eta_{\mu\nu}+h_{\mu\nu}(x)\tag{1}$$ ...
5
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1answer
287 views

Why is there no Green's function when no gauge is chosen (Linearized Gravity)

I'm working on linearized gravity, and have come to the point where you need to choose a gauge to simplify the Einstein equation for the perturbative field. Reading a paper, I came across the claim "...
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1answer
272 views

Gravitational waves (linearized gravity) [closed]

Even in the Schwarzschild metric we can write $g_{uv}=\eta_{uv}+h_{uv}$ where $h_{uv}$ is very small. So after some coordinate transformation (using gauge freedom) we can simplify the Einstein ...
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0answers
335 views

Linearized Einstein equation on a general background metric

All of my texts only give the Linearized Einstein equation on the Minkowski background so I thought I'd try and figure it out by hand today. Using the standard perturbation $h_{\mu\nu}$ and denoting ...
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1answer
143 views

Gravitational wave equations worked out

Is there a website where gravitational wave equations are worked out numerically? I would like to experiment with mass configurations but can't find examples.
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0answers
135 views

What is the explicit form of $\tau^{\alpha\beta}$ in the linearized Einstein field equations $\Box h^{\alpha\beta}=-16\pi\tau^{\alpha\beta}$?

If we let $h^{\alpha\beta}=\eta^{\alpha\beta}-g^{\alpha\beta}\sqrt{|det(g)|}$ then, according to wikipedia, the Einstein Field Equations become $$\Box h^{\alpha\beta}=-16\pi\tau^{\alpha\beta},$$ where ...
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1answer
351 views

Lagrangian mechanics - small oscillations around equilibrium diagonalization

In my analytical mechanics class, we have been taught that normal modes of small oscillations around equilibrium are given by the solution of $$ p(\omega) = \det(K-\omega^2M) = 0 $$ Where $K_{ij} = -\...