1
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2answers
78 views

Stress-energy-momentum tensor

In Wald's General Relativity, he writes on pg 61 For an observer with 4-velocity $v^a$, the component $T_{ab}v^a v^b$ is interpreted as the energy density, i.e. the mass-energy per unit volume, as ...
1
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2answers
55 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? ...
2
votes
1answer
86 views

Stress energy tensor and the covariant derivative of the 4-momentum

Another basic question. I have usually seen the stress energy tensor $T^{ij}$ described as the flow of the 4-momentum field $p^i$ along direction $x^j$ in spacetime with $p^0$ as energy and $x^0$ as ...
0
votes
1answer
93 views

Stress-energy tensor explicitly in terms of the metric tensor

I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $R_{\mu\nu}$ and scalar curvature ...
2
votes
1answer
59 views

Energy-momentum tensor for dust

We all know that the energy-momentum tensor for dust is just $T^{\alpha\beta}=\rho_0v^\alpha v^\beta,$ where $\rho_0$ is the mass density in the dust's rest frame and $v^α$ is the dust's ...
3
votes
2answers
94 views

Temperature as frequency spectrum of stress-energy tensor?

I am currently learning general relativity, and in the textbooks that I am reading, temperature seems to be treated as a scalar field, extraneous to the geometry of spacetime. This is puzzling me, ...
0
votes
1answer
89 views

Physical interpretation of $Q^i = \partial _\nu T^{i \nu}$

I'm trouble with exercise 1.8 of Carroll's Space-Time and Geometry: If $\partial_\nu T^{\mu \nu} = Q^\mu$, what physically does the spatial vector $Q^i$ represent? Use the dust energy momentum ...
0
votes
0answers
40 views

What is the “momentum” referred to in the energy-momentum tensor

What is the "momentum" referred to in the energy momentum tensor from GR? Is it $m\dot{x}$ or is it the canonical momentum $\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}}\right)$ Also, I ...
1
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0answers
51 views

Will accelerating a massive particle generates a blackhole? [duplicate]

I have a naive question about blackhole. If I accelerate a massive particle very close to the speed of light, the particle will have large energy-momentum tensor. Will it become a blackhole?
1
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2answers
195 views

Einstein equation and scalar field stress-energy tensor

Let's have interaction between gravitational and scalar real fields. For an action of gravitational field in vacuum I add term $S_{m} = \int d^{4}x\sqrt{-g}L_{m}$, where $$ L_{m} = \frac{1}{2}g^{\mu ...
5
votes
1answer
388 views

How to find the Stress-Energy tensor?

I am a bit at loss about how to proceed to find the stress-energy tensor given some distribution of matter. The Wikipedia page gives some examples, and some (inequivalent) definitions for it: Using ...
2
votes
1answer
71 views

Why is general relativity only formulated in continuum terms?

So, when we are discussing Newtonian mechanics, we treat particles as point particles. In continuum mechanics, which I understand to be a version in which mass is continuously distributed, we have ...
0
votes
1answer
119 views

Stress-energy tensor. Why this general form?

How is the stress energy tensor obtained? In most textbooks, it's simply stated as $$T^\mu{}_\nu=(\rho+P)U^\mu U_\nu-P\delta^\mu{}_\nu$$ I can see why this makes sense for a comoving observer at ...
1
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3answers
282 views

Is it true to say *Space time curvature* $\Leftrightarrow$ *Matter*

Is it true to say Space time curvature and Matter are just the same thing, part of the same coin and that therefore Space time curvature $\Leftrightarrow$ Matter? In other words is Space time ...
8
votes
1answer
235 views

How does one get these definitions of the energy momentum tensor?

I was just reading a book - Mirror Symmetry by Clay Mathematics Institute, and on Page 402 of the book, the writer says that energy momentum tensor is defined classically by $$\delta S = -\frac{1}{4 ...
3
votes
1answer
318 views

energy momentum tensor and covariant derivative

In field theory, the energy momentum defined as the functional derivative wrt the metric $T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}$ (up to a sign depending on ...
0
votes
1answer
121 views

Can inertial mass affect gravity of the object? [duplicate]

Every time I watch this TV program that discusses about all the facts about the universe , and it came to a point where they said that as an object approaches the speed of light the mass of the object ...
1
vote
1answer
210 views

Static Spherical symmetric solution of Einstein's equations with a perfect fluid

I am reading Wald for the interior solutions of a static spherical metric. Assume it to be of the form $$ds^2 = -f(r)dt^2 + h(r)dr^2 + r^2 ( d{\theta^2} \sin^2{\theta}d{\phi^2})$$ Wald states: For a ...
2
votes
1answer
125 views

Angular momentum for the Kerr solution of a rotating blackhole

I am reading 't Hooft's noted on Black holes, where he quotes the Kerr metric for a black hole rotating about the z-axis as follows: He later says: "The parameter a can be identified with the ...
6
votes
1answer
422 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 ...
3
votes
1answer
106 views

Have general relativistic effects of all of the components of the stress-energy tensor been measured?

The stress-energy tensor is: Have general relativisic effects of all of the components of the stress-energy tensor been measured? I've heard that the accelerating expansion of the universe is due to ...
0
votes
1answer
128 views

Does the actual curvature of spacetime hold energy?

My understanding of GR is that curvature of spacetime reflects the density of energy-matter. Does the curvature itself have energy? Or if energy is assigned to curvature it simply reflects the energy ...
2
votes
1answer
920 views

Stress energy tensor of a perfect fluid and four-velocity

In the following demonstration, there is an error, but I cannot find where. (I explicitely put the $c^2$ to keep track of units). We consider a metric $g_{\mu\nu}$ with a signature $(-, +, +, +)$ : ...
3
votes
1answer
164 views

Sign crazyness on the stress energy tensor?

I would like to know on what depends the sign of the stress energy tensor in the following formula : $T_{\mu\nu}=\pm(\rho c^2+P)u_{\mu}u_{\nu} \pm P g_{\mu\nu}$ In my case the metric is equal to ...
5
votes
1answer
270 views

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 $E^2 = (pc)^2 + (mc^2)^2$ ...
1
vote
1answer
303 views

Scalar field stress energy tensor

Can anyone explain why $T_{\mu \nu} = \frac{2}{\sqrt{-g}} \frac{\delta \mathcal{L}_M}{\delta g^{\mu \nu}} $, other than justifying it from the einstein field equations?
2
votes
1answer
530 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 ...
6
votes
2answers
274 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 $\rho$ ...
4
votes
1answer
820 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?
5
votes
1answer
224 views

Confused about indices of the Ricci tensor

In an intro to GR book the Ricci tensor is given as: $$R_{\mu\nu}=\partial_{\lambda}\Gamma_{\mu \nu}^{\lambda}-\Gamma_{\lambda \sigma}^{\lambda}\Gamma_{\mu \nu}^{\sigma}-[\partial_{\nu}\Gamma_{\mu ...
3
votes
3answers
197 views

Having trouble seeing the similarity between these two energy-momentum tensors

Leonard Suskind gives the following formulation of the energy-momentum tensor in his Stanford lectures on GR (#10, I believe): $$T_{\mu \nu}=\partial_{\mu}\phi \partial_{\nu}\phi-\frac{1}{2}g_{\mu ...
1
vote
2answers
312 views

fourth rank tensor for stress energy

The Weyl tensor equates the Riemann tensor in vacuum $$ C_{\mu \nu \eta \lambda} = R_{\mu \nu \eta \lambda} $$ So it makes me wonder about the tensor $$T_{\mu \nu \eta \lambda} = C_{\mu \nu \eta ...
3
votes
4answers
322 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). ...
6
votes
1answer
865 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 ...
1
vote
2answers
256 views

Finding the correct units for the energy-momentum tensor?

I'm trying to understand the energy-momentum tensor $T^{\mu\nu}$ but I'm confused about the units. My textbook says the components of $T^{\mu\nu}$ are $\mathrm{Jm^{-3}}$. Four-momentum is is given ...
5
votes
1answer
272 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 ...
4
votes
1answer
121 views

General parameters of the stress energy tensor in local inertial frame

A general 4x4 symmetric tensor has 10 independent components. How many components are we free to prescribe in the local inertial frame? For example, relativistic dust is $\mbox{diag}(\rho c^2, 0, 0, ...
3
votes
3answers
1k views

Action principle for the Electromagnetism and Gravity

Here is the formula for the stress energy tensor: $$ T_{\mu\nu} = - {2\over\sqrt{ |\det g| }}{\delta S_{EM}\over \delta g^{\mu\nu}} $$ (This follows from varying the total action $S ...