1
vote
1answer
103 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 ...
1
vote
1answer
63 views

Ricci curvature tensor, definition of symbols

So I know that $$R_{μν}:=R^λ_{μλν}$$ is the Ricci curvature tensor (where $R^λ_{μλν}$ is the Riemann Tensor). This is in Einstein's field equations: $$R_{μν}-\frac{1}{2}g_{μν}R=\frac{8πG}{c^4}Τ_{μν}$$ ...
1
vote
0answers
51 views

Variation of the purely covariant Riemann tensor

I need to find the variation of the purely covariant Riemann tensor with respect to the metric $g^{\mu \nu}$, i.e. $\delta R_{\rho \sigma \mu \nu}$. I know that, $R_{\rho \sigma \mu \nu} = g_{\rho ...
1
vote
2answers
93 views

Planetary motion: integration of equation of motion

I was reading Planetary Motion (page 117) in Barry Spain's Tensor calculus, and stupidly enough, I didn't understand this. The equations are : $$\frac{d^2\psi}{d\sigma^2} + ...
2
votes
2answers
76 views

Covariant derivative of a covariant tensor wrt superscript

Is it true that when you take the covariant derivative of a covariant tensor, do you always have to do with a subscript? What if you do it wrt a superscript?Does the first term (with the partial ...
0
votes
1answer
87 views

Square of a tensor

I think, $$\sigma_{ij}\sigma^{ij} = \sigma^2.$$ However, on the Wikipedia page on Raychaudhuri equation, It was mentioned: $$\sigma^2=\frac{1}{2}\sigma^{ij}\sigma_{ij}$$ I am confused, but I think ...
5
votes
2answers
287 views

Perturbation metric problem

I know this is an already answered question, but I couldn't make head or tail of it, and it's bugging me. I know I'm probably asking a silly question, but please bear with me as I'm 14 and this is my ...
2
votes
1answer
148 views

The Spin Connection

Why do we need to introduce the spin connection coefficients $\omega_{\mu \space \space b}^{\space \space a} $ in General Relativity? To me, they just look (mathematically) like the Christoffel ...
0
votes
1answer
61 views

Covariant derivative of a vanishing tensor component [closed]

Is the covariant derivative of a vanishing tensor component necessarily zero?
0
votes
0answers
68 views

Einstein frame vs. Matter frame

What is the difference between Einstein frame and Matter frame in General Relativity? -A brief comment on each could be useful too. These two frames were used in this manuscript ...
2
votes
1answer
175 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
134 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
116 views

Hypersurface Normal

Could anyone explain why $$n^{a}n_{a}=\pm1$$ where $n^{a}$ is the normal to the hypersurface
1
vote
2answers
112 views

Inner products in relativity

In physics, the definition of a dot (inner) product is often between a vector (“contravariant vector”) and a covector (“covariant vector”). However, in mathematics, a dot product is always defined ...
12
votes
4answers
452 views

Is the concept of tensor rank useful in physics?

The term 'tensor rank' is sporadically used in the mathematical literature to denote the minimum number of simple terms (i.e. tensor products of vectors) needed to express the tensor. This is ...
1
vote
0answers
54 views

Contracting Indices in General relativity [duplicate]

I was reading a book about general relativity and I came across these two equations $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{\Gamma}^{\mu}_{\sigma\rho}+ ...
9
votes
1answer
311 views

Contracting Indices

Does anyone know how to get from (1) to (2) in the system $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{{\Gamma}}^{\mu}_{\sigma\rho}+ ...
0
votes
1answer
93 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
2answers
95 views

What does this summation mean in relativity?

Equation 1.2 of 't Hooft's Introduction to General Relativity gives the Lorentz transformations: $$ (x^\mu)' = \sum\limits_{\nu = 1}^4 {L^\mu}_\nu x^\nu $$ Is this the sum of four square matrices ...
2
votes
1answer
451 views

Deriving an equation involving Killing vectors

I'm currently studying Carroll's GR book Spacetime & Geometry, and ran into some trouble understanding the text. When discussing Killing vectors, Carroll mentions that one can derive ...
3
votes
1answer
202 views

Riemann tensor notation and Christoffel symbol notation

In paper by Barnich and Brandt Covariant theory of asymptotic symmetries, conservation laws and central charges they defined the Riemann tensor like this: $$R_{\rho\mu\nu}^{\quad \ \ ...
1
vote
1answer
135 views

Transformation rule of a partial derivative

We know the following transformation rule: $$ \partial'_b = \frac{\partial}{\partial x'^b} = \frac{\partial x^c}{\partial x'^b} \, \frac{\partial}{\partial x^c} = \frac{\partial x^c}{\partial x'^b} ...
2
votes
2answers
260 views

Notation for anti-symmetric part of a tensor

I know that $A_{[a} B_{b]} = \frac{1}{2!}(A_{a}B_{b} - A_{b}B_{a})$ But how can write $E_{[a} F_{bc]}$ like the above? Can you provide a reference where this notational matter is discussed?
2
votes
1answer
266 views

Weight of a tensor density

Is there any freedom in choosing the weight of a tensor density? I have seen in some papers that they introduce a tensor density made from metric with a special weight. There is a tensor density with ...
16
votes
2answers
371 views

In relativity, can/should every measurement be reduced to measuring a scalar?

Different authors seem to attach different levels of importance to keeping track of the exact tensor valences of various physical quantities. In the strict-Catholic-school-nun camp, we have Burke ...
3
votes
1answer
1k views

Riemann tensor in 2d and 3d

Ok so I seem to be missing something here. I know that the number of independent coefficients of the Riemann tensor is $\frac{1}{12} n^2 (n^2-1)$, which means in 2d it's 1 (i.e. Riemann tensor given ...
1
vote
2answers
235 views

How do you show from the index notation that the change of frame formula for a metric must involve the transpose?

Let $x^\mu$ and $x^{'\mu}$ be two coordinate systems related by $$dx^{'\mu}~=~S^\mu{}_\nu~ dx^\mu.$$ In index notation the metric in both systems are related by: ...
1
vote
1answer
206 views

Weyl & Riemann curvature tensors and gravitational “physical” quantities in Einstein vacuum equations

If we look at the Einstein vacuum equations, that is without matter (there is the possibility or curvature without matter), for instance we may consider gravitational waves. The question is: Is there ...
-1
votes
1answer
215 views

Metric tensor in General Relativity or otherwise [closed]

What is the metric tensor? How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices? How it relates to distance function (metric) and angles? How ...
2
votes
2answers
141 views

Tensors in general relativity

This is a question on the nitty-gritty bits of general relativity. Would anybody mind teaching me how to work these indices? Definitions: Throughout the following, repeated indices are to be summed ...
0
votes
1answer
66 views

Why is it suffice to show Tensorial identity on a tensor composed of two vectors?

I've encounter many proves of Tensorail identity that begin with assuming our tensor can be written in form of: $T^{\alpha\beta}=u^{\alpha}v^{\beta}$ . As helpful is it might be, I'm not sure if its ...
12
votes
1answer
430 views

What are the local covariant tensors one can form from the metric?

Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain ...
2
votes
1answer
81 views

General expression of the redshift: explanation?

In some papers, authors put the following formula for the cosmological redshift $z$ : $1+z=\frac{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{S}}{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{O}}$ where : $S$ ...
4
votes
0answers
242 views

Tensor equations in General Relativity

In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent. Question: why is this true? I'm looking for a ...
3
votes
0answers
119 views

Curvature and spacetime

Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
5
votes
1answer
257 views

When a variation of a tensor is not a tensor?

In a comment about variation of metric tensor it was shown that $$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$ which is contrary to the usual rule of lowering indeces of a ...
2
votes
1answer
1k 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
191 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 ...
1
vote
0answers
123 views

Lecture Notes confusion: Constructing the Einstein Equation

This question is on the construction of the Einstein Field Equation. In my notes, it is said that The most general form of the Ricci tensor $R_{ab}$ is $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ ...
0
votes
1answer
47 views

Zero-zero (lower indicies) term for affine connection ($\Gamma_{00}^\lambda$), why do some terms dissapear?

More simply a tensor algebra question, but in General relativity I have the following when I calculate $\Gamma_{00}^\lambda$:- $$ \Gamma_{00}^\lambda = \frac{1}{2}g^{\nu\lambda}\left( \frac{\partial ...
2
votes
1answer
425 views

Ricci identity/Riemann curvature tensor and covectors

Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
2
votes
1answer
263 views

Contracting the Riemann tensor issues, p540 hobson

I am stuck trying to work through something on p540 in Hobson (General Relativity: An Introduction for Physicists), one is supposed to use the variation of the full Riemann tensor and then contract it ...
2
votes
1answer
93 views

Non-diagonal elements when switching metric signature?

Considering a metric tensor with the signature $(-,+,+,+)$: $g_{\mu\nu}= \begin{pmatrix} -c^2 & g_{01} & g_{02} & g_{03}\\ g_{10} & a^2 & g_{12} & g_{13}\\ g_{20} & g_{21} ...
1
vote
0answers
114 views

Einstein +Maxwell 's tensor

Why is it true that we can deduce that Einstein's GR equations coupled with Maxwell's EM equations may be written in the form $$R_{ij}=C(F_{ik}F_j^{\,\,k}-{1\over 4}g_{ij}F_{mn}F^{mn})$$ without ...
1
vote
1answer
166 views

Relationship between a formal vector derivative and time evolution of an operator

I'm an undergraduate in physics, with all the lack of knowledge inherent in that. In two of my classes, my professors introduced two equations which look eerily similar. The first, from general ...
2
votes
1answer
486 views

Difference between $\partial$ and $\nabla$ in general relativity

I read a lot in Road to Reality, so I think I might use some general relativity terms where I should only special ones. In our lectures we just had $\partial_\mu$ which would have the plain partial ...
8
votes
1answer
613 views

Diffeomorphisms, Isometries And General Relativity

Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while. Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ ...
5
votes
1answer
233 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 ...
6
votes
3answers
183 views

From Manifold to Manifold?

Tensor equations are supposed to stay invariant in form wrt coordinate transformations where the metric is preserved. It is important to take note of the fact that invariance in form of the tensor ...