The variables used in general relativity to describe the shape of spacetime. If your question is about metric units, use the tag "units", and/or "si-units" if it is about the SI system specifically.

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1answer
106 views

Cosmological metric with off-diagonal terms?

In the context of Cosmology models, What are examples of metrics with off-diagonal terms?
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0answers
65 views

Angular Momentum with Upper Index

I am asked to show $[L^2,L_i] = 0 $, but with the definition : $L^2 \equiv L_i L^i$ I tried this: $[L_i L^i,L_i] = L_i [L^i,L_i] + [L_i,L_i]L^i$ We know that : $[L_i,L_i]$ = 0 , so we have, $[L_i ...
0
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1answer
414 views

General Relativity: Christoffel symbol identity

I want to show that $$\Gamma ^{\mu}_{\mu \nu}=\partial _\nu (\ln \sqrt{|g|}) .$$ (Here $|g|$ denotes the determinant of the metric.) Working out the left hand side:\begin{align} \Gamma ^{\mu}_{\mu ...
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2answers
389 views

General relativity: Induced metric and Killing vector fields

Assume that in spacetime ($M,g_{ab}$) there is a hypersurface generated by a set of independent one-parameter transformations acting on one single point, the generators of these transformations being ...
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1answer
30 views

Flat space current conservation sign confusion

It is said that in Minkowski spacetime, the current conservation law for the number current $N^\mu$ where $N^0$ is the number density and $N^i, i=1,2,3$ is the particle flux in the $x^i $ direction, ...
3
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0answers
95 views

About Dirac equation in curved spacetime (spherical)

I would like to ask you about the separation of variables of the Dirac equation in curved space-time. The metric is given by $$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+\alpha^{2}r^{2}\sin^{2}\theta ...
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3answers
187 views

Coordinate transformations of the metric tensor

Let's have metric (it describes the space-time of uniformly accelerating observer in Minkowski space-time) $$ ds^2 = v^2du^2 - dv^2. \qquad (.0) $$ I need to find expressions for $u = f(x, t), v = ...
2
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1answer
218 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 ...
2
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2answers
128 views

Distinguish between Past and Future

When writing the metric in Minkowski space, how can we distinguish between the past and the future? I understand the answer after drawing the light cone but I want to know how we get that by just ...
2
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1answer
186 views

Lightcone coordinates

The Light cone coordinates are defined as $$x^+ = x^0 + x^3$$ $$x^- = x^0 - x^3$$ Then in the light cone coordinates the position 4-vector becomes: $(x^+, x^-, x^1, x^2)$ . Zwiebach in his A First ...
3
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3answers
589 views

How the Lorentz transformation affects the metric tensor?

After performing a Lorentz transformation, the orthogonal coordinates will become askew, as in the following figure: and in such coordinate system, according to this Wikipedia article, the metric ...
3
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1answer
117 views

Index raising and lowering - how does it work?

In the context of four-dimensional spacetime, how does the metric turn a tangent vector into a gradient, and vice versa? By this I mean that I know the metric can be used to raise and lower indices: ...
2
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1answer
335 views

How to derive the metric for a 2-sphere

I have a question in Polchinski's string theory vol I p 167. It is said For example, $$ds^2= \frac{ 4 r^2 dz d \bar{z} }{(1+ z \bar{z})^2} = \frac{ 4 r^2 du d\bar{u}}{ (1+ z \bar{z})^2} ...
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1answer
193 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 ...
1
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1answer
50 views

A question about variation of metric under Weyl and coordinate transformations

I have a question about deriving variation of metric under Weyl and coordinate transformations in Polchinski's string theory (3.3.16). Under transformation $$\zeta: g \rightarrow g^{\zeta}, \,\,\, ...
3
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1answer
314 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 ...
4
votes
1answer
111 views

Are third derivatives of metric perturbations zero?

I'm working on a problem related to fluid perturbations of stars. I'm following this paper. They start with the Einstein equation: $$G_{\alpha \beta} = 8 \pi G T_{\alpha \beta}$$ and then perturb the ...
1
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2answers
191 views

Derivation of freely falling frame in Schwarzschild spacetime

Thinking about the equivalence principle, is there a nice, simple way to show that a local, freely falling frame in Schwarzschild spacetime is described by the Minkowski metric ...
1
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1answer
307 views

Schwarzschild solution

I am calculating for many hours and I am really confused with this exercise. Consider a comoving observer sitting at constant spatial coordinates$(r∗,θ∗,φ*)$, around a Schwarzschild black hole of ...
0
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1answer
83 views

What physical sense has following transformation?

Let's have an interval expression for Friedmann Universe with 3-metric of a sphere, $$ ds^{2} = c^{2}dt^{2} - c^{2}\frac{ch^{2}(Ht)}{H^{2}}\left( d\rho^{2} + sin^{2}(\rho )(d\theta^{2} + ...
9
votes
3answers
326 views

Equation of motion of a photon in a given metric

I have this metric: $$ds^2=-dt^2+e^tdx^2$$ and I want to find the equation of motion (of x). for that i thought I have two options: using E.L. with the Lagrangian: $L=-\dot t ^2+e^t\dot x ^2 $. ...
11
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2answers
401 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
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1answer
78 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 : $S$ ...
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1answer
173 views

Finding the Basis vectors of a Killing field vector space

I have solved the Killing vector equations for a 2-sphere and got the following answer. $A,B,C$ are three integration constants as expected. $$\xi_{\theta}=A \sin{\phi}+B\cos{\phi}$$ ...
6
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2answers
182 views

What's the basic premise of General Relativity?

What is the basic assumption(s) required to explore general relativity? For example, if one merely assumes that the speed of light $c$ is the same for all observers, and the laws of physics are the ...
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1answer
139 views

Vector analysis in curvilinear coordinates using the metric tensor

In Weinberg's Gravitation, the formula for the volume element in curviliniar coordinates is given by $$dV=h_1 h_2 h_3 dx^1 dx^2 dx^3.$$ The metric is given by $ds^2=h_1^2 dx_1^2+h_2^2 ...
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2answers
305 views

Kronecker delta confusion

I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same): ...
3
votes
1answer
104 views

Would this be a metric?

would a matrix $M$ with diagonal entries not necessarily equal 1, i.e. diag $M = (a,1,1,1)$ be a metric if $a \neq 1$ or $\neq 0$? I.e. in this case would this be like some sort of more general ...
1
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1answer
152 views

Evaluating the Ricci tensor effectively

If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
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0answers
50 views

metric extension outside the light cone

Could anyone explain what "extending the solution" beyond the past light cone means? Say, for example, if I have a metric (no coordinate singularities), how can I extend it to the outside of the past ...
3
votes
1answer
141 views

Why does the Kruskal diagram extend to all 4 quadrants?

Why is it that the Kruskal diagram is always seen extended to all 4 quadrants when the definitions of the $U,V$ coordinates don't seem to suggest that the coordinates are not defined in, say, the 3rd ...
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1answer
241 views

What is the Lorentz tensor with a superscript and subscript index?

I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form: $\Lambda^{\mu}_{\nu} X^{\nu}$. I am moderately familiar ...
0
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1answer
683 views

Christoffel symbol for Schwarzschild metric

I know that the christoffel (second kind) can be defined like this: $$\Gamma^m_{ij} = \frac{1}{2} g^{mk}(\frac{\partial g_{ki}}{\partial U^j}+\frac{\partial g_{jk}}{\partial U^i}-\frac{\partial ...
3
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1answer
384 views

Derivation of the volume element (which uses the metric tensor)?

I have often seen $\sqrt{-g}$ in integrals, especially actions, where $g=\mathrm{det}(g_{\mu \nu})$. Does anyone know of a derivation that shows that this is indeed the volume element which must be ...
4
votes
1answer
215 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 ...
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1answer
269 views

Why vary the action with respect to the inverse metric?

Whenever I have read texts which employ actions that contain metric tensors, such as the Nambu-Goto, Polyakov or Einstein-Hilbert action, the equations of motion are derived by varying with respect to ...
2
votes
1answer
208 views

Parallel transport of a vector along a closed curve in curvilinear coordinates

There is an expression indicating the change of the vector parallel translation along a closed infinitesimal curve in curvilinear coordinates (one way of introducing curvature tensor): $$ \Delta A_{k} ...
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0answers
71 views

Null vector fields given Bondi metric

I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric $g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$ with $d\Omega$-standard metric ...
2
votes
1answer
108 views

Plane waves in QFT

Suppose we work in the metric $(-1,+1)$. How do we describe an incoming particle with a plane wave; $\exp(-\mathrm ikx)$ or $\exp(+\mathrm ikx)$? What's the difference? Does it change if we work in ...
3
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1answer
182 views

The most general form of the metric for a homogeneous, isotropic and static space-time

What is the most general form of the metric for a homogeneous, isotropic and static space-time? For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) ...
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0answers
59 views

Singularities in Schwarzchild space-time

Can anyone explain when a co-ordinate and geometric singularity arise in Schwarzschild space-time with the element $$ ...
2
votes
2answers
332 views

Ricci tensor for a 3-sphere without Math packets

Let's have the metric for a 3-sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's ...
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0answers
478 views

How to calculate Riemann and Ricci tensors for a sphere? [closed]

Let's have the metric for a sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's ...
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0answers
44 views

How to prove the derive the expression for space part of Riemann tensor for homogeneous and isotropic space-time?

It's not a homework!! For spheric, hyperbolic and flat case $$ dl^{2} = R^{2}\left(d \psi^{2} + sin^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})\right), $$ $$ dl^{2} = R^{2}\left(d ...
1
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1answer
270 views

Contraction of the metric tensor

This is perhaps a simple tensor calculus problem -- but I just can't see why... I have notes (in GR) that contains a proof of the statement In space of constant sectional curvature, $K$ is ...
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0answers
110 views

The interior of a cylinder as an Einstein manifold

The interior of a curved cylinder is an Einstein manifold (the Ricci Curvature Tensor is proportional to the Metric $R_{\mu\nu}=kg_{\mu\nu}$) since it has a constant curvature. However, I was unable ...
3
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3answers
385 views

How scalar curvature of following spacetime can be equal to zero?

For an interval of this spacetime, $$ ds^{2} = c^{2}dt^{2} - c^{2}t^{2}(d \psi^{2} + sh^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})), $$ scalar curvature is equal to zero. Also, Ricci ...
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1answer
58 views

Change of variables in an interval expression

This question is a continuation of How to calculate a scalar curvature fast? . Let's have Lorentz-Fock spacetime with an interval $$ d \hat {s}^{2} = \frac{t_{0}^{2}R^{2}}{\hat {t}^{4}}\left( d \hat ...
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3answers
172 views

Combining metric tensors/curvature tensors

I was thinking about the following scenario: Consider a particle which causes a metric $g_{\mu\nu}$ on an otherwise Minkowski spacetime (or any manifold). Now, consider another particle, somewhere in ...
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2answers
775 views

What is metric of spherical coordinates $(t,r,\theta,\phi)$?

In spherical coordinates the flat space-time metric takes: $$ds^2=-c^2dt^2+dr^2+r^2d\Omega^2$$ where $r^2d\Omega^2$ come from when the signature of metric $g_{\mu\nu}$ is (-,+,+,+)? what is ...