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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Does non-mass-energy generate a gravitational field?

At a very basic level I know that gravity isn't generated by mass but rather the stress-energy tensor and when I wave my hands a lot it seems like that implies that energy in $E^2 = (pc)^2 + (mc^2)^2$ ...
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917 views

The derivation of the Belinfante-Rosenfeld tensor

It seems me that there is a "difference" (at least apparently) in how the Belinfante-Rosenfeld tensor is thought of in section 7.4 of Volume 1 of Weinberg's QFT book and in section 2.5.1 of the ...
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2answers
398 views

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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2answers
3k views

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
896 views

Potential Energy in General Relativity

I often hear about how general relativity is very complicated because of all forms of energy are considered, including gravitation's own gravitational binding energy. I have two questions: In ...
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1answer
824 views

Why is the stress-energy tensor symmetric?

The relativistic stress-energy tensor $T$ is important in both special and general relativity. Why is it symmetric, with $T_{\mu\nu}=T_{\nu\mu}$? As a secondary question, how does this relate to the ...
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1k views

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 ...
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903 views

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 ...
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129 views

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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1answer
499 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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3answers
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Why is the (nonrelativistic) stress tensor linear and symmetric?

From wikipedia: "...the stress vector $T$ across a surface will always be a linear function of the surface's normal vector $n$, the unit-length vector that is perpendicular to it. ...The linear ...
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1answer
283 views

Showing symmetry of the stress tensor by applying divergence theorem to $\int\int_{\delta V(t)} \vec{x}\times \vec{t} dS$

I'm currently working through the symmetry of the stress tensor, in relation to viscous flow. I am looking at this by examining the conservation of angular momentum equation for a material volume ...
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2answers
127 views

What are the factors affecting the spacetime curvature?

Large masses in space as stars and planets cause a curvature in the spacetime fabric. What are the factors that affect this curvature? Is it only mass? And can we conclude these factors using Tensors? ...
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1answer
328 views

Energy-Momentum Tensor in QFT vs. GR

What is the correspondence between the conserved canonical energy-momentum tensor, which is $$ T^{\mu\nu}_{can} := \sum_{i=1}^N\frac{\delta\mathcal{L}_{Matter}}{\delta(\partial_\mu f_i)}\partial^\nu ...
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3answers
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Energy momentum tensor from Noether's theorem

in the book Quantum Field Theory by Itzykson and Zuber the following derivation for the stress-energy tensor is proposed (p.22): Assume a Lagrangian density depending on the spacetime coordinates $x$ ...
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2answers
285 views

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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1answer
397 views

Energy-momentum tensor of Bosonic Ghost Action in String Theory

When quantizing bosonic string theory by means of the path integral, one inverts the Faddeev-Popov determinant by going to Grassmann variables, yielding: $$ S_{\mathrm{ghosts}} = ...
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1answer
565 views

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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0answers
58 views

Is there a general formula to translate from *canonical* to *physical* momentum?

In Peskin and Schroeder, after having derived a conserved tensor $T^{\mu \nu}$ associated with translations in space-time (the stress-energy tensor), it is said that the charges $\int d^3 x T^{0i}$: ...
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0answers
125 views

Derivation of correction to canonical stress energy tensor due to addition of total divergence to Lagrangian

It is mentioned in almost every text book that equations of motions are not modified if we add a total divergence of some vector $$\partial_\mu \ X^{\mu}$$ to Lagrangian but canonical stress energy ...
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69 views

Constructing conserved current given the lagrangian

Consider the following Lagrangian for a massive vector field $A_{\mu}$ in Euclidean space time: $$\mathcal L = \frac{1}{4} F^{\alpha \beta}F_{\alpha \beta} + \frac{1}{2}m^2 A^{\alpha}A_{\alpha}$$ ...
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1answer
446 views

Source term of the Einstein field equation

My copy of Feynman's "Six Not-So-Easy Pieces" has an interesting introduction by Roger Penrose. In that introduction (copyright 1997 according to the copyright page), Penrose complains that Feynman's ...
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2answers
162 views

Could Dust Equation of State have some negative pressure?

Traditionally the cosmological equation of state of cold matter (so-called dust) is simply: $$p = 0.$$ But, in Newtonian terms, each particle is gravitationally attracting every other particle. ...
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4answers
392 views

Formulation of general relativity

EDIT: I think I can pinpoint my confusion a bit better. Here comes my updated question (I'm not sure what the standard way of doing things is - please let me know if I should delete the old version). ...
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1answer
104 views

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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101 views

Constructing Ward identity associated with conserved currents

Consider constructing the Ward identity associated with Lorentz invariance. It is possible to find a 3rd rank tensor $B^{\rho \mu \nu}$ antisymmetric in the first two indices, then the stress-energy ...
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1answer
440 views

Symmetry of stress-energy tensor

Why in the general case of classical field theory canonical stress-energy tensor doesn't have symmetry of the permutation of the indices? For explanation, let's have a "derivation" of an expression ...
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1answer
503 views

Why is the Maxwell Stress Tensor symmetric?

What is the physical meaning of the Maxwell Stress tensor symmetry?
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1answer
1k views

What is the stress energy tensor?

I'm trying to understand the Einstein Field equation equipped only with training in Riemannian geometry. My question is very simple although I cant extract the answer from the wikipedia page: Is the ...
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1answer
2k views

Flow of momentum is pressure

In the diagonal terms of the energy-momentum tensor, the flow of $x$-momentum in the $x$-direction is the $x$-pressure. Why the flow of momentum is pressure?
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1answer
66 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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0answers
31 views

Do anyone know a good software that where I can easily find the metric from the stress-energy tensor? [duplicate]

I'm using SageMath but the obtainment of the metric from the stress-energy tensor is not trivial, i.e., it is not implemented in a predefined function. Do anyone know a good software that where I can ...
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1answer
52 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 ...