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.

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For the Schwarzschild metric, are the values of the Ricci tensor and Ricci scalars always zero?

If we use the Schwarzschild metric to solve the Einstein field equations, would the values of the Ricci tensor and scalars always be zero?
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Expectation value of stress-energy tensor

In the following page, notes by \textit{Brout}, how they are calculating 2.86 first line to second line?
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The divergence of the stress-energy tensor vanishes; is this statement sufficient to derive the Einstein field equations?

Can one derive the Einstein field equations from this statement alone? $$0 = T^{\mu\nu}{}_{;\nu} = \nabla_\nu T^{\mu\nu} = T^{\mu\nu}{}_{,\nu} + \Gamma^{\mu}{}_{\sigma\nu}T^{\sigma\nu} + \Gamma^{\nu}{...
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Physical Situations of Specific Stress Energy Tensor?

I'm having trouble picturing what the physical situation of a non-symmetric stress would be. Say I have a stress tensor $T_{ab} = \begin{pmatrix} T_{00} & T_{01}& 0 & 0 \\ 0&T_{11}&...
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Derivative of the Lagrangian with respect to the metric tensor

I'm trying to calculate the derivative of the Lagrangian $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi-\frac{1}{2}m^2\phi^2$$ with respect to the metric tensor $g_{\mu\nu}$, with the ...
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Could the curvature of spacetime, as in general relativity, result from the interaction of quantum fields?

If both the general and special theories of relativity deal with space as spacetime, then the special theory of relativity deals with spacetime as flat, and the general theory of relativity deals with ...
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Energy-momentum tensor for the k-essence theory

could anyone please explain or show some simple steps how using matter action: $S = \int d^4x \sqrt{-g} L(X, \phi)$, where $X = \frac{1}{2} g^{\mu \nu} \nabla_\mu \phi \nabla_\nu \phi$ We can derive ...
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Is a stress tensor still symmetric when the object is rotating?

I am trying to simulate a spinning & flying deformable football with FEM method. It is always accelerating, instead of keeping static. Let an undeformed nodal position on this football $X \in \...
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Raising the index of the stress energy tensor [closed]

By considering the variation of a lagrangian which has no explicit space time dependence under an arbitrary spacetime translation, $a^{\mu}$ I've seen that we can express the variation in the ...
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How to see that the electromagnetic stress-energy tensor satisfies the null energy condition?

I am trying to show that the Maxwell stress-energy tensor, $$T_{\mu\nu} = \frac{1}{4\pi}\left( F_{\mu\rho} F^{\rho}{}_{\nu} - \frac{1}{4}\eta_{\mu\nu}F_{\rho \sigma} F^{\rho\sigma} \right),$$ ...
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Finding the stress tensor in the Stokes equation from free energy

I have found free energy for a system of particles. The free energy is a functional of a scalar field which is the area of the particles. So I have the following free energy $F(A,p)= \int dx dy \: f[A(...
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How can any spatially extended object have 4-momentum assigned?

We know that the 4-momentum of a point particle is of the form $p^{\mu} = (E/c, p^{i})$, whose transformations across different inertial reference frames are given by Lorentz transformations. However, ...
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Pressure in stokes flow

In the stokes equation for an incompressible fluid, there is a pressure term that enforces the fluid to have $\nabla \cdot u=0$, where $u$ is the velocity field. The stokes equation reads: $$\eta \...
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Can the Lorentz force equation in curved spacetime be derived from the Einstein-Maxwell equations?

Given the Einstein field equations, $$R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} = \kappa T_{\mu\nu}$$ that imply in particular that $\nabla_\mu T^{\mu\nu}=0$, one can show, using the explicit form of $T^{\...
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Linear Momentum in General Relativity

My question is, does a particle moving in a straight line at constant velocity through empty space create "frame dragging" that would tend to entrain other bodies in the direction of its ...
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Does the stress energy tensor scale with the metric tensor?

Question I had some thoughts from a previous question of mine. If I have a metric $g^{\mu \nu}$ $$g^{\mu \nu} \to \lambda g^{\mu \nu}$$ Then does it automatically follow for the stress energy tensor $...
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Energy-Momentum tensor in the non-relativistic limit of Klein-Gordon Field

Assume we have a real Klein Gordon field $\phi(x,y,z,t)$, and we do the non-relativistic expansion of it in terms of a complex field $\psi(x,y,z,t)$ $$\phi=\frac{1}{\sqrt{2m}}(\psi e^{-imt}+\psi^* e^{...
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Why for 6d SCFT we consider (1,0) and (2,0) only?

It is known to have a stress-energy tensor we must have the supercharges $\mathcal{N}<2$. My confusion is why in most cases we consider chiral supercharges, and what is the problem with $\mathcal{N}...
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Lower vs Upper indices in stress energy tensor

In Goldstein Classical Mechanics, chapter 13 page 56, equations 13.30, the canonical stress energy tensor $T_\mu^{\,\,\,\nu}$ is defiend as: $$T_\mu^{\,\,\,\nu}=\frac{\partial\mathcal{L}}{\partial \...
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Do Einstein field equations only relate local spacetime curvature to local energy-momentum of matter?

Do Einstein field equations only relate local spacetime curvature to local energy-momentum of matter? If so, can we extend Einstein field equations globally relating global spacetime curvature to ...
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Understanding EFE: RHS linear, LHS not?

Einstein's field equations are nonlinear. That means it is not allowed to add up the metric tensors. However, on the RHS of the field equations, there is only the stress-energy-momentum tensor, and it ...
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Energy-momentum tensor of a perfect fluid flowing at the speed of light?

The general energy-momentum tensor for a fluid is given by (working with $c=1$ convention) $$\begin{equation}\label{T}\tag{1}T^{\mu\nu}=\left(\rho+p\right)U^\mu U^\nu+p g^{\mu\nu}\end{equation}$$ with ...
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Covariant vs. contravariant definition of the Energy-Momentum tensor

I have a question regarding the definition of the energy-momentum tensor. I've seen it defined as a (2,0) tensor, so it has 2 upper indices $T^{ab}$, but many times it is written as a (0,2) tensor ...
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Show Energy-momentum operator transforms as a tensor under Lorentz transformations

I know, from my professor notes, that a general field operator can transform under Lorentz as $$U(\Lambda)\hat{\mathcal{O}}^r(x)U^\dagger(\Lambda)=\hat{\cal{O}}^{r'} {M(\Lambda)_{r'}}^r$$ where $M(\...
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Noether/Hilbert energy-momentum tensor

In chapter 4 of Carroll's book Spacetime and geometry he finds using the Hilbert action that the energy-momentum tensor for a scalar field is (see eq. (4.79)) $$T_{\mu\nu}^{\phi}=\nabla_\mu\phi\nabla_\...
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Difference between a stress energy tensor and an effective stress energy tensor?

As I was working through this article by Bronnikov regarding Einstein–Cartan theory: Bronnikov, K. A., & Galiakhmetov, A. M. (2015). Wormholes without exotic matter in Einstein–Cartan theory. ...
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In QFTCS, is the indeterminacy in the local energy density due to vacuum particle-antiparticle creation?

In QFT in curved spacetime, there is an indeterminacy in the local energy density (because of the indeterminacy in defining annihilation/creation operator) if the spacetime is not stationary. Is it ...
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Interpretation of Angular Momentum flux from stress energy tensor in Black hole superradiance

While studying the superradiance of a scalar field in Kerr geometry, we show that the energy and angular momentum of the Kerr black hole are extracted by the superradiant modes. I understand the ...
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Stress tensor trace anomaly in two dimensions

I'm trying to calculate the expectation value of the stress tensor in 2D following the book "Quantum fields in curved space" (Birrell and Davies). In 2D the divergent contribution to the one-...
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Stress tensor for a real massive vector field in General Relativity

Let's consider the classical Lagrangian density for a real vector field $A_\mu$, $$ \mathcal{L}_v=\sqrt{-g}\left(-\frac{1}{2}A_{\mu;\alpha}A^{\mu; \alpha}-\frac{1}{2} R_{\mu \nu} A^\mu A^\nu+\frac{1}{...
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Why is viscous stress a tensorial quantity?

In an incompressible fluid, the viscous stress (in Cartesian) is defined by \begin{align} \tau_{ij} = \eta(\partial_i v_j + \partial_j v_i) \end{align} for dynamic viscosity $\eta$ and velocity field $...
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Null vectors that aren't the zero vector in general relativity?

So I was trying to understand the null energy condition of $T_{μν}k^μk^ν≥0$ Where $k$ is an "arbitrary future-directed null vector" and couldn't really wrap my head around how the $k$ is ...
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Radiation Pressure derivation

Radiation pressures mathematical expression according to Wikipedia is, $\frac{1}{\mu_0 c}\vec{E} × \vec{B}$ "Radiation pressure is the mechanical pressure(force/area) exerted upon any surface due ...
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How to interpret the Lagrangian $\int d^3x \sqrt {-g} R$?

In GR, one of the degrees of freedom is that of the metric tensor. That means a tensor $g_{ab}$ at each point of space. The other degree of freedom is that of the energy distribution. That's also a ...
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EMT in 2D Euclidean Yang Mills

In pure 2D Euclidean YM theory with $SO(8)$ gauge group. For Lagrangian $\frac{1}{4g^2} Tr(F_{\mu\nu}F^{\mu\nu})$ is energy momentum tensor $$T^{\mu\nu}=\frac{1}{g} Tr(-F^{\mu\rho}F_\rho^\nu+1/4\eta^{...
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Symmetric form of Einstein Field Equations?

So normally, taking $c = 1$ and ${8\pi G = 1}$, and assuming the cosmological constant is negligible, the Einstein field equations read: $$R_{\mu \nu} - \frac{1}2Rg_{\mu\nu} = T_{\mu \nu}.$$ However, ...
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What curves spacetime in Schwarzschild metric? [duplicate]

I understand that the Schwarzschild solution is valid in the outside region of a massive object, with no other masses involved. Therefore the energy-momentum tensor is 0. But then: what curves space? ...
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Energy Momentum Tensor for massive Real Scalar Field in GR with metric signature (+ - - -) [duplicate]

I wanted to calculate the Energy Momentum Tensor for the massive real scalar field in Kerr Background. But instead of the usual (- + + +) metric signature, I have to use the (+ - - -) signature for ...
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Is there a reason that the stress-energy-momentum tensor curves the spacetime?

The famous field equations of Albert Einstein describe, how spacetime is curved by the stress-energy-momentum tensor. The spacetime is curved by matter and energy. General relativity is the underlying ...
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Role of Maxwell's stress tensor in the conservation of momentum of an EM wave

Defining the Wave: Let's assume an electromagnetic wave exists in the form $\vec{E}(x,y,z,t) = \vec{E}_0 \, cos(kx-\omega t)$ $\vec{B}(x,y,z,t) = \vec{B}_0 \, cos(kx-\omega t)$ How the wave interacts ...
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Relativistic energy density of a fluid in different frames

I have a fluid in a local inertial frame with stress-energy tensor $T^{\mu\nu}$. In another local inertial frame that is moving with velocity $u$ with respect to the first one, it is said that the ...
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Gravity's self-energy

Suppose we have a single massive point particle. In the absence of "potentials", the content of the stress-energy tensor would be dictated uniquely by the particle's mass and trajectory (...
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Confusion regarding 4-Velocity Derivative Identity (for conservation of energy momentum tensor) in Carroll's Spacetime and Geometry

During Carroll's discussion of the energy-momentum tensor for a perfect fluid (page 36), he writes out that its divergence should be zero. He then expands this as follows: $$\partial_\mu T^{\mu\nu} = \...
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Will an object with linear velocity in outer space experience frame dragging effects?

Will an object moving non-accelerated in outer space experience frame-dragging? It seems the mass contributions to the strdss-energy-momentum tensor are distributed spherically symmetrical, and don't ...
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Stress energy tensor components for a perfect fluid

The stress energy tensor for a perfect fluid is given by $$T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}-pg_{\mu\nu}$$ where U is the 4-velocity. The matrix components of the SEM are written as $$T_{\mu\nu}=diag(\...
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Why is the stress tensor contravariant?

The stress tensor relates the traction $\vec{t}$ (force per area) on a surface with surface normal $\vec{n}$ usually written as (when disregarding co- and contravariance) $$ t_j = \sigma_{ij} n_i.$$ ...
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How do we know what definition to choose for the stress energy tensor for a situation?

Consider the Einstein Field equations $$R_{\mu\nu}-\frac12Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}.$$ Typically, the stress energy tensor $T_{\mu\nu}$ is assumed to be a perfect fluid, which implies ...
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How to find the energy-momentum tensor of a free relativistic particle from its lagrangian?

Consider a free relativistic particle in Minkowski spacetime. Its standard action is the following, where $\sigma$ is an arbitrary parametrization ($\tau$ is the particle's proper time. I'm using ...
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Why this boundary term could be ignored for a free relativistic particle?

How can we justify that the boundary integral we get from the following could be ignored, when we want to find the equation of motion? I consider the energy-momentum of a free particle in special ...
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Symmetric stress-energy tensor in CFT

I'm a bit confused reading about the stress-energy tensor and conformal Ward identities in Di Francesco. My question is in a similar spirit to this one from several years ago, but the question was not ...
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