Questions tagged [stress-energy-momentum-tensor]

A rank-2 tensor in relativity, which expresses the flux of energy-momentum along timelike and spacelike axes. Also known as the energy-momentum tensor. In the Einstein field equations, it is the source of gravitational fields.

Filter by
Sorted by
Tagged with
1
vote
1answer
24 views

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 ...
0
votes
0answers
9 views

How to prove symmetry of stress tensor using torque equation? [closed]

** You have to write the angular momentum equation i.e. Torque = (Moment of Inertia)x(Angular Acceleration) for an infinitesimally small 3D element (about any axis of your choice). Finally, you have ...
0
votes
1answer
29 views

Why is there an extra term in definition of Noether current for spacetime translations?

I am reading Schwartz's Quantum Field Theory textbook. In chapter 3, Schwartz first defines the conserved current for a symmetry $\phi \rightarrow \phi + \delta \phi$ that depends on a parameter $\...
0
votes
1answer
28 views

Calculation of thermal expansion

how are the thermal expansion of a solid and the stress tensor related? \begin{equation} \int_{\mathbb{R}^{3N}}\frac{\Delta E_p}{\Delta V}\rho_{H_h}(u)du=\frac{2}{3V}\int_{\mathbb{R}^{3N}}E_p\rho_{...
0
votes
1answer
26 views

Can the Lagrangian density of a field be derived from the stress-energy tensor?

I have been learning some field theory and learning about Lagrangian and Hamiltonian density. In classical mechanics, the Hamiltonian is the energy of the system in terms of position and momentum. I ...
2
votes
1answer
51 views

Expectation value of $T_{\mu\nu}$ for uniform accelerating motion

Birrell and Davies in their book state that if $\langle \psi|T_{\mu\nu}|\psi\rangle = 0$ in one reference frame then it must be zero for all others whereas the particle content may vary for different ...
0
votes
0answers
15 views

Do the spatial integrals of the EM stress tensor with all sources transform as a 4-vector over all space?

Jackson writes in section 16.4 of Classical Electrodynamics: So far our discussion of the Abraham-Lorentz model of a classical charged particle has been nonrelativistic, with apologies for the ...
0
votes
1answer
55 views

Prove that the value of the cosmological constant equals the energy density of the vacuum

I know that Einstein introduced his cosmological constant assuming it as an independent parameter, something characteristic of the Universe, in itself, but the term of it in the field equations can be ...
0
votes
2answers
65 views

Deriving energy-momentum tensor in Schwartz QFT

I don't understand how to derive the chain of equalities right after equation (3.31): We are told $$\frac{\delta \phi}{\delta \xi^\nu} = \partial_\nu \phi\tag{3.30}$$ and $$\frac{\delta \mathcal{L}}{\...
5
votes
1answer
82 views

Stress-Energy Tensor of Reissner-Nordstrom solution

I am trying to derive the R-N solution and i am following Blau's notes (to be found here http://www.blau.itp.unibe.ch/newlecturesGR.pdf) pages 677-679. With the same metric ansatz: $$ ds^2 = -A(r)dt^2 ...
2
votes
0answers
35 views

Improved stress-energy tensor derivation from Green's function

In chapter 4 of the book Quantum fields in curved space by Birrell and Davies, the authors provide a derivation of the Casimir energy for a massless scalar field (in 4-D) using the method of images ...
1
vote
1answer
39 views

FLRW Coupling to Perfect Fluid

In order to obtain the Friedmann equations from the Lagrangian formalism, as far as I understand, one way is to minimally couple a scalar field $\phi(t)$ to the FLRW metric, i.e. $S=\int d^4 x \sqrt{-...
1
vote
2answers
41 views

4-momentum through a spacelike-othogonal hypersurface for a perfect fluid in special relativity

My understanding of the stress-energy tensor in special relativity (or in general relativity), is that it gives you the flux density of 4-momentum flowing through an oriented 3D hypersurface. So at ...
3
votes
1answer
106 views

Conservation of Electromagnetic Energy-Momentum Tensor in GR

In the follow Planck units have been used. The Electromagnetic Energy-Momentum Tensor is $T^{\mu\nu} = \frac{1}{4 \pi} \left[ F^{\mu \alpha}F^\nu{}_{\alpha} - \frac{1}{4} g^{\mu\nu}F_{\alpha\beta} F^{...
3
votes
1answer
60 views

Stress tensor of fluid in equilibrium, inertial frame

There are some points in this wikipedia chapter. Main equation is: $$ T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta} $$ where $c$ is explicit. The one ...
2
votes
2answers
106 views

Lorentz covariance properties of energy and momentum of electromagnetic field

In special relativity the 4-tuple of numbers, energy and 3 components of momentum, form a 4-vector with respect to Lorentz transformations. Is it true that the analogous 4-tuple of energy and ...
0
votes
1answer
68 views

Proof of Cauchy's Lemma

This question is related to this one I asked some time ago: Reason for Symmetry of stress tensors. The reason behind the symmetry of the Cauchy stress tensor seems to be Cauchy's Lemma. Once you ...
1
vote
1answer
77 views

How to count degrees of freedom in a symmetric $N \times N$ matrix?

I am reading Wayne Hu's short lecture on cosmology mathematical infrastructure (https://arxiv.org/abs/astro-ph/0402060), and have several questions. Some background for us lazy people that don't want ...
1
vote
1answer
36 views

Relationship between stress-energy tensor for a point particle and its Lagrangian

The Lagrangian for a (relativistic) point particle with rest mass $m$ and velocity $v$ is: $$L=-\frac{m}{\gamma (v)}$$ (using $c=1$). Over on Wikipedia we can find the Stress-energy tensor for said ...
0
votes
1answer
64 views

Is this energy equation from Einstein's book the energy used in the stress-energy tensor for gravity?

Einstein in his book: Relativity: The Special and General Theory. http://www.ibiblio.org/ebooks/Einstein/Einstein_Relativity.pdf [edit] on page 56, I believe] writes, "the expression for energy&...
0
votes
1answer
41 views

Show that modified energy-momentum tensor defines conserved currents

In David Tong's notes on quantum field theory, one of the problems asks to show that a tensor defined by $$\Theta^{\mu\nu}=T^{\mu\nu}-F^{\rho\mu}\partial_\rho A^\nu$$ where $F^{\rho\mu}=\partial^\rho ...
0
votes
1answer
106 views

How to use general expression for Noether's current to get the energy-momentum conservation law?

The most general form of the Noether's current (see here and here) is given by $$j^\mu(x)=\sum\limits_a\frac{\partial \mathscr{L}}{\partial(\partial_\mu\phi_a)}\delta\phi_a -\theta^{\mu\nu}\delta x_\...
1
vote
2answers
99 views

How to derive this $\dfrac{dT}{d\tau}$?

I am studying the paper "Gravitational field of a particle falling in a Scharzschild geometry analyzed in tensor harmonics" by Zerilli. The author calculates the gravitational radiation ...
0
votes
2answers
61 views

Geometry constraining the Stress-energy tensor?

Question Let's say I have a metric in radial coordinates such that at $r \to \infty$ we find flat spacetime: $$ds^2 \sim -c^2 dt^2 + dr^2 + r^2 d \Omega^2$$ where $ds^2$ is the line element and $t$ is ...
4
votes
0answers
65 views

Degrees of freedom in quantum mechanics and $c$-theorem

In 2D there is notion of degrees of freedom. D.o.f. defined from correlation functions of stress-enegry tensor: $$ F(|z|^2) = z^4 \langle T_{zz}(z,\bar{z}) T_{zz}(0,0) \rangle \\ G (|z|^2) = \frac{z^...
1
vote
1answer
50 views

Special Relativity: Interpretation of the partial derivate of Stress-Energy Tensor

This question is based on Carroll's book Spacetime and Geometry, specifically from page 33 to page 36. In the upper mentioned section we define the Stress-Energy Tensor as: The flux of the four ...
0
votes
0answers
52 views

How to calculate the attractive force in a gravitational field in General Relativity?

I believe that a gravitational field has energy, as Weinburg wrote in his textbook Gravitation and Cosmology, on page 171: "... ... the gravitational field does carry energy and momentum." ...
3
votes
1answer
105 views

Proving an Identity “easily” as written by author, in GR electromagnetism [closed]

The last day and, some days before, I found myself incapable of proving an Equation, while the author said it was "easily" deducted (Chapter 6, Page 127, 3+1 Ideal Magnetohydrodynamics- Éric ...
3
votes
2answers
173 views

Solutions to Einstein Field Equations where $T_{\mu \nu} = 0$

My Level/Background: I have just completed my first year of undergrad. In high school, I completed AP Physics C Mechanics and Electricity and Magnetism. In my first year of undergrad, I completed a ...
1
vote
1answer
48 views

Are energy and momentum imposed by purely geometrical properties of spacetime?

If we defined spacetime as a purely geometrical (not physical) structure of the kind that is in general relativity (a 4-dimensional Lorentzian manifold), would it automatically have properties that ...
1
vote
1answer
42 views

Reasonable ways to couple matter with metric

In general relativity without matter, the equation of motion of the metric field is described by the Hilbert's action or the Einstein tensor $G$. It's natural to lead to this conclusion once one ...
0
votes
0answers
27 views

Energy density of gravitational field [duplicate]

Can we define the energy density of gravity field? If so, what is the formula, and the newtonian approximation? In newtonian mechanics, we can easily obtain the gravitational potential energy. But if ...
0
votes
0answers
20 views

Help on Hamiltonian tensor equations in Einstein's original General Relativity papers

Studying Einstein's original Die Grundlage der allgemeinen Relativitätstheorie published in 1916's Annalen Der Physik, I came across Equations 47b) regarding the gravity contribution to the stress-...
1
vote
1answer
44 views

Help on electromagnetic tensor equations in Einstein's original General Relativity papers

Studying Einstein's original Die Grundlage der allgemeinen Relativitätstheorie published in 1916's Annalen Der Physik, I came across Equations 66) and 66a) regarding the electromagnetic contribution ...
0
votes
1answer
72 views

Can a stress-energy tensor induce signature changes on the metric?

Suppose we use the signature of a Riemannian manifold $$ \eta^{\mu\nu}=\operatorname{diag}(+,+,+,+) $$ as the starting point to describe a 4d Euclidean version of general relativity. Alternatively one ...
1
vote
0answers
26 views

Sound wave in relativistic perfect fluid

I was trying to solve the following problem from the problem book on relativty: Problem 5.22. Show that the velocity of sound $v_{s}$ in a relativistic perfect fluid is given by $$v_{s}^{2}=\partial p ...
0
votes
1answer
31 views

Maxwell stress tensor for electric field

If we need to calculate the time-averaged Maxwell stress tensor for an arbitrary field like $$ \vec{E}=E_{0x}e^{ikz-iwt}\hat{i} + E_{0y}e^{ikz-iwt}\hat{j}+E_{0z}e^{ikz-iwt}\hat{k} $$ I know we should ...
0
votes
0answers
37 views

Energy-Momentum Tensor not traceless?

In my university, we have a different approach for deriving the energy-momentum tensor for electromagnetic field in vacuum. The result is: $T_i {_j} = \epsilon_0 \ (\frac{1}{2}E^2 \delta_i {_j}\ - ...
3
votes
2answers
142 views

Does the Landau-Lifshitz pseudotensor contain all non-linear terms in $h_{\mu\nu}$?

I was recently reading up some literature on gravitational waves in curved spaces and came across the following confusion. Basically starting from the Einstein Field Equation (EFE), $$G_{\mu\nu} = 8\...
2
votes
1answer
43 views

Does Energy-momentum tensor from Polyakov action have anything to do with the energy and momentum of the string?

I'm a bit confused with the idea of EMT from the Polyakov action. The EMT is derived by variation of the action with respect to the metric which provide the constraint to the theory, $T_{ab}=0$. ...
1
vote
1answer
77 views

Physical interpretation of the following stress-energy tensor : $T^{\mu\nu}=X^\mu X^\nu$

I have come across the following stress-energy tensor and I was wondering if anyone know of a physical system this could correspond to? $$ T^{\mu\nu}=\pmatrix{(X^0)^2 & X^0X^1 & X^0X^2&X^...
1
vote
0answers
44 views

Dilaton background in Replica wormholes publication

Refering to the paper "Replica Wormholes and the Entropy of Hawking Radiation" by Almheiri et al. in arXiv:1991.12333. The authors consider Jackiw-Teitelboim gravity theory describing a ...
0
votes
1answer
35 views

General Relativity - Curvature of two superimposed perfect pressureless fluids

If we imagine a hypothetical perfect fluid with no pressure, then in its inertial rest frame its stress-energy tensor would only have one component: $\quad T_{00} = \rho$ Solving the field equation ...
1
vote
1answer
54 views

Doubt on Tetrads, Energy-momentum tensors and Einstein's equations

Given, for instance, the perfect fluid energy-momentum tensor: $$T_{\mu\nu} = (\rho+p)u_{\mu}u_{\nu} - pg_{\mu\nu}\tag{1}$$ We can put (due to diagonalization procedure) into the diagonal for as: $$...
0
votes
0answers
31 views

EM Energy-momentum tensor

I'm trying to prove the following equation which is recieved by the energy momentum tensor: $$\partial_{\mu} T^{\mu \nu}=\frac{1}{c}j_{\mu}F^{\mu \nu}. $$ The energy momentum tensor is defined by $$ ...
3
votes
0answers
160 views

Relativistic rotation

Consider the following two scenarios: a) A particle is rotating in the $x-y$ plane about some point fixed in lab frame at a radius $a$ with relativistic angular speed $\omega$. Do not include the ...
0
votes
2answers
101 views

Rotational motion and special relativity

I was trying to solve two questions from problem book on relativity and gravitation by lightman.Questions are Calculate the nonzero components in an inertial frame S of the stress-energy tensor for ...
4
votes
1answer
149 views

In layman's terms why is the time component of the stress-energy tensor associated with energy density?

Why is the time component of the stress energy tensor $T^{00}$ associated as energy density?
3
votes
1answer
60 views

RG fixed points and $T_{\mu\nu}$

It is common to refer to fixed points of the renormalization group as scale invariant theories. This statement can be formulated as $$ \beta(\mu) \Big |_{\mu^*} = 0 \; \; \Longrightarrow \; \; T^{\mu}...
4
votes
2answers
104 views

What is a coordinate-free formulation of deformation theory?

For example how are stress, strain and shear tensors described invariantly, without any coordinates, purely in a geometric manner? A formulation that avoids indices coordinates and matrices, even in ...

1
2 3 4 5
15