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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The significance of the pressure term within the momentum-energy tensor [duplicate]

EDIT: this question is based around my notion regarding the possible role of potential energy in the momentum energy tensor T$_{\mu\nu}$, The answer below resolves the question and I have deleted ...
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Intuitive understanding of the elements in the stress-energy tensor

There is an image in the Wikipedia about the stress-energy tensor: I have a rough understanding of the stress tensor: you imagine cutting out a tiny cube from the fluid and form a matrix out of the ...
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50 views

The divergence of the Stress Energy Tensor

I have been studying general relativity and I have often seen in textbooks that the divergence of the stress energy tensor is zero. $$T^{\mu\nu}_{;\nu} = 0$$ but is it possible to contract and ...
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1answer
270 views

What does Weinberg–Witten theorem want to express?

Weinberg-Witten theorem states that massless particles (either composite or elementary) with spin $j > 1/2$ cannot carry a Lorentz-covariant current, while massless particles with spin $j > 1$ ...
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1answer
466 views

Traceless of stress-energy tensor in $d=2$

This is a question regarding Francesco, section 4.3.3. In this section, he considers the two-point function $$ S_{\mu\nu\rho\sigma}(x) = \left< T_{\mu\nu}(x) T_{\rho\sigma}(0)\right> $$ He then ...
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Stress Force - Understanding Cauchy Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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3answers
769 views

Maxwell's Stress Tensor

What really is the Maxwell Stress Tensor? I understand that it's derived from $$\mathbf {F} = \int _V ( \mathbf E + \mathbf v \times \mathbf B )\rho \ d \tau$$ Griffiths describes this as "total EM ...
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1answer
43 views

Converging light

Imagine that we emit a light pulse. As is the nature of light, it will expand. However, light is affected by gravitational fields and light has its own. Therefore, will the light converge given ...
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45 views

What is the gravitational effect inside a electromagnetic shield due to an external electromagnetic field?

I am new in General Relativity. I know that electromagnetic field (or, the electromagnetic energy tensor, $T^{ik}=1/4\pi[1/4F_{mn}F^{mn}g^{ik}-F^i_lF^{lk}]$) can affect gravitation. Now if we take a ...
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1answer
37 views

Derivation of the energy-momentum tensor for an imperfect fluid

In chapter 7 of the "Physical Foundations of Cosmology" Mukhanov uses this energy-momentum tensor for an imperfect fluid: $T^\mu_\nu = (\rho + p)u^\mu u_\nu - p\delta^\mu_\nu - \eta(P^\mu_\gamma ...
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59 views

Deriving the energy-momentum tensor conservation equation in complex coordinates, Polchinski 2.4.2

I am trying to derive equation (2.4.2) in Polchinski's string theory textbook, $$\overline \partial T_{zz}=\partial T_{\overline z \overline z} = 0 \tag{2.4.2}.$$ Using the conservation equation, ...
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54 views

Why can we not choose the stress tensor in a CFT to be identically symmetric?

The stress tensor for a conformal field theory (or any quantum field theory) can be derived from the action $S$ by the functional derivative $$T^{\mu \nu} ~=~ -\frac{2}{\sqrt{|g|}}\frac{\delta ...
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220 views

Pressure and density using a general Lagrangian

Given a lagrangian of a form: \begin{equation}\mathcal{L}=f(\phi,\partial_{\mu}\phi\partial^{\mu}\phi)\end{equation} where $f$ is a function, I need to derive pressure and density in a FLRW universe ...
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Calculate energy-density of known plasma in microwave?

Let's assume I want to create a plasma in a regular household microwave similar to this home-made experiment. Although I am dealing with a small amount of mass, I assume that the addition of microwave ...
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2answers
101 views

Pass to globally conserved currents from locally conserved currents in curved spacetime

Let us begin with a Lagrangian of the form $$\mathscr L= \frac 12 \sqrt{-g}g^{\mu\nu}\partial_\mu\phi(x)\partial_\nu\phi(x)+\mathscr L_g,$$ where $$\mathscr L_g=\frac 1{16\pi k}\sqrt{-g}R.$$ ...
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Does this identity that applies to the metric tensor also apply to the stress-energy tensor?

Okay so if the $g_{00}$ component of the metric is $-c^2$ and $g_{11}=g_{22}=g_{33}$ and all the other other components are zero, the question is simple, would similar identities apply to the ...
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electroweak field contribution to the space curvature in GR

i've just found out that EM stress energy tensor along with gravitational stress energy contribute to the curvature of space. So, does the electroweak field also contribute to the curvature of space? ...
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1answer
68 views

In general, can a Lagrangian density depend on space-time explicitly?

In an exercise on classical field theories, I'm trying to derive the general formula of the Energy-momentum tensor. According to the formula in the lecture notes, this tensor includes a term of minus ...
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5answers
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Negative pressure, tension, and energy conditions

We have lots of common everyday experience with positive pressure, the canonical example is a gas. But other examples of positive pressure are easy to imagine: for instance, a solid that gets ...
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39 views

What observation(s)--if any--confirm that the types & concentrations of energy, which are influenced by gravity, are the same ones that cause gravity?

General relativity allows various forms of energy to participate in the gravitational force. What observation(s)--if any--confirm general relativity's notion that the various types & ...
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65 views

Real-world evidence that non-massive entities (or even: antiparticles), and their behaviors, are sources of gravity?

The theory of general relativity tells us that non-massive entities, and their behaviors, are possible sources of gravity. Mass isn't needed, the theory says. What's the real-world evidence that ...
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Finding the metric tensor from the Einstein field equation?

I have have set my self a challenge to learn all the maths behind the Einstein field equation (EFE), and from reading it seems that the Metric tensor is the thing we are trying to find (from the 10 ...
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Step by step algorithm to solve Einstein's equations

I cannot completely understand what is a regular method to solve Einstein's equations in GR when there are no handy hints like spherical symmetry or time-independence. E.g. how can one derive ...
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3answers
366 views

Einstein tensor in Friedmann equations : where is the missing $c^2$?

I would like to demonstrate the several forms of the Friedmann equations WITH the $c^2$ factors. Everything is fine ... apart that I have a missing $c^2$ factor somewhere. In all the following $\rho$ ...
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4answers
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What is the meaning of Einstein's field equation in terms of source and its effects on curvature?

The Einstein's Field Equation is $$R_{\mu\nu}-(1/2)g_{\mu\nu}R=-8\pi T_{\mu\nu},$$ where the left hand side is the curvature term and the right hand side is the source term (see, Hartle). Now, in the ...
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How can I use Einstein's field equations to find the metric tensor? [duplicate]

I have watched and read a lot on the topic of General Relativity and the geometry behind it. I am confident that I can derive an approximation of the the stress-energy-momentum tensor with just the ...
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1answer
100 views

Is any spacetime metric physically realizable?

Given a spacetime metric, you could work out a stress-energy tensor for each position that would result in that metric. I know building a wormhole requires negative energy densities, which are ...
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58 views

Will a stress-energy tensor have the same identities as it's metric?

Say I have a metric tensor where $$g_{00} = -c^{2}\ and $$ $$g_{01}=g_{02}=g_{03}=0$$ and $$g_{12}=g_{13}=g_{23}$$ and $$g_{11}=g_{22}=g_{33}$$ My question is straightforward: would the same or ...
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70 views

Time dilation as an effect of energy density

Has any relation been observed or postulated to exist between the energy-density (or the surrounding space) of an object and time dilation? i.e. Higher energy density==>Slower rate of time?
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55 views

Hamiltonian density of classical Klein-Gordon field

I am working my way through Peskin and Schroeder section 2.2 and trying to show that $T^{00}$ is equivalent to the expression $\frac{1}{2}\pi^2-\frac{1}{2}(\nabla \phi)^2-\frac{1}{2}m^2\phi^2$ in ...
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65 views

Computation of $T^{\mu\nu}$ from Lagrangian density $\mathscr{L} $

I am trying to understand how upper and lower indices are connected when computing the energy-momentum tensor. In particular, I found the simple problem where the Lagrangian density is given as ...
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43 views

Energy-Momentum Tensor with mixed indices

I know that $T_{\mu\nu}$ is the Energy-Momentum Tensor and $T=g^{\mu\nu}T_{\mu\nu}$, but does anyone know what $T^{\nu}_{\mu}$ is and how its calculated?
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Different forms of the Einstein field equation

I am working my way through the wonderfully written introduction "General relativity for mathematicians" by Sachs & Wu. I am indeed a mathematics student and find this book to be well suited to ...
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1answer
38 views

Stress Energy tensor for non-relativistic particle [closed]

I am trying to write down the elements of stress-energy tensor for a point particle moving with non-relativistic velocity $v$. I have written : $ T_{00} = mc\delta(\vec{r} -\vec{r}') $ and $ T_{0i} ...
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1answer
452 views

Why does the Ricci tensor vanishes in Schwarzschild metric? [duplicate]

If the Schwarzschild metric is suppose to describe the behaviour of a spherical object in flat space, so the Schwarzschild is different from the flat metric because it describes curved space so why ...
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1answer
102 views

Doubt regarding stress-energy tensor definition

I'm having some trouble understanding the following definition of the stress energy tensor: $T^{\mu\nu}$ is the flux of four-momentum $p^{\mu}$ across a surface of constant $x^{\nu}$. Here's an ...
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Is energy-momentum of curvature a boundary/holographic density?

Since the beginnings of General Relativity, we have had this awkward, unholy separation of the universe in marble versus wood. divergence of the stress-energy momentum holds at all points of ...
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Ideal, isotropic fluid and stress tensor

An ideal fluid is the one which cannot support any shearing stress. It also doesn't have viscosity. My question is what does it mean by a fluid to be isotropic? Is an ideal fluid necessarily isotropic ...
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Definition of stress at the microscale

Take, for simplicity, a Lennard-Jones fluid below the critical temperature, which is to say that there is a phase separation into fluid and gas and thus an interface is formed. The macroscale picture ...
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624 views

Dirac field and stress-energy tensor density

I read somewhere that stress-energy tensor density is a symmetric tensor. But if I take the Dirac Field tensor: $$T^{\mu \nu}=i \psi^\dagger \gamma^0 \gamma^\mu \partial^\nu \psi $$ How could I ...
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1answer
51 views

how to increase the moment of inertia of a hollow aluminium pipe without changing the outer diameter [closed]

how to increase the moment of inertia of a hollow aluminium pipe with external diameter fixed and only allowed to change the shape of internal section for example rectangular hole or extruded section ...
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1answer
66 views

Stress-Energy Tensor Integral Identity [closed]

I'm attempting to work a problem in Schutz's A First Course in General Relativity, and I'm running into something curious with tensor indices. The problem states: Use the Identity ...
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Symmetric energy-momentum tensor using derivative wrt. metric

I can find the Noether current for space time translation symmetry by demanding that the first order correction to the Lagrangian vanishes upon infinitesimal translations of coordinates. But in cases ...
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2answers
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Why is gravity such a unique force?

My knowledge on this particular field of physics is very sketchy, but I frequently hear of a theoretical "graviton", the quantum of the gravitational field. So I guess most physicists' assumption is ...
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1answer
55 views

Why does Weyl invariance imply a traceless energy-momentum tensor?

I've begun to self-study String Theory from Polchinski and Becker, Becker and Schwarz. I don't see why the fact that the Polyakov action is invariant under Weyl transformations is related to the ...
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Stress-energy tensor for a fermionic Lagrangian in curved spacetime - which one appears in the EFE?

So, suppose I have an action of the type: $$ S =\int \text{d}^4 x\sqrt{-g}( \frac{i}{2} (\bar{\psi} \gamma_\mu \nabla^\mu\psi - \nabla^\mu\bar{\psi} \gamma_\mu \psi) +\alpha \bar{\psi} \gamma_\mu ...
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On the isotropy of materials

Good morning. I am working on Honeycomb structures and first of all I would like to understand whether it is Isotropic or not, and , if the latter holds which kind of anisotropy it has. How to do it? ...
6
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3answers
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Is it in general true that $\nabla_\mu T^{\mu\nu}=0$ implies the matter equations of motion?

I know of several cases where the covariant conservation of the energy momentum tensor $\nabla_\mu T^{\mu\nu}=0$ can be used to derive the equations of motion of the matter fields. Is this in general ...
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Symmetrizing the Canonical Energy-Momentum Tensor

The Canonical energy momentum tensor is given by $$T_{\mu\nu} = \frac{\partial {\cal L}}{\partial (\partial^\mu \phi_s)} \partial_\nu \phi_s - g_{\mu\nu} {\cal L} $$ A priori, there is no reason to ...
10
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2answers
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Symmetry of the $3\times 3$ Cauchy Stress Tensor

When presenting the stress tensor (say in a non-relativistic context), it is shown to be a tensor in the sense that it is a linear vector transformation: it operates on a vector $n$ (the normal to a ...