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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Once I calculate the Riemann curvature tensor, what do I do with it?

I am considering the Schwarzschild metric. I have calculated my Christoffel symbols and am able to calculate the Riemann tensor (I think). In short, I have done a bunch of work to find this thing ...
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2answers
103 views

Sign convention for the Minkowski metric $\eta_{\mu\nu}$

In special relativity, one is confronted with a quadratic form called proper time, which is $c^2t^2-(x^2+y^2+z^2)$, $t$ being time and $x,y,z$ being the space coordinates. One usually introduces a ...
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0answers
54 views

Gauge invariant quantities

In the context of cosmological perturbation one write the most general perturbed metric as $$ ...
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3answers
43 views

Definition of non-degenerate metric tensor

We know that a metric has a property which is called non-degeneracy. I was searching for what does that mean and saw it associated with the fact that $det(g_{\mu\nu})\neq0$. How does this relate to ...
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1answer
71 views

How does the Lorentz boost change if we introduce transformation to the minkowski metric

Let's say we have the Lorentz boost given by the $ \Lambda^\mu_\nu$ in the Minkowski metric $diag\{1,-1,-1,-1\}$. Now if I do a transformation on the Minkowski metric such that the new metric is ...
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2answers
71 views

Off-diagonal terms in metric for 4D space-time [closed]

Consider a delta between two events in 4D space-time written as a 4-vector, $x^\mu=(dt, dR)$. The time $dt$ is a scalar difference in time. The 3-vector $dR$ points some direction in space. One ...
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3answers
703 views

Gödel's solutions to Einstein's relativity equations and their consequences

Gödel gave certain solutions to Einstein's relativity equations that involved a rotating universe or something unusual like that; that predicted stable wormholes could exist and therefore time travel, ...
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3answers
462 views

Space is expanding so what is time doing? [duplicate]

Space is expanding and as we know space and time are intrinsically linked to be now known as spacetime. What is happening to time during expansion? Is there more time, longer time or is the time part ...
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4answers
94 views

Normal Vectors to these Hypersurfaces on a Lorentzian Manifold

With respect to the coordinates $(x^{0},x^{1},x^{2},x^{3})=(v,r,\theta,\phi)$, we have the following components of the metric tensor: $\begin{bmatrix} g_{00} & g_{01} & g_{02} & ...
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1answer
41 views

Weyl scalar calculation

I'm trying to compute Weyl scalars, but don't really understand the formulae for them, in the sense I don't understand how to compute them. Let's take ...
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1answer
15 views

Commutativity and symmetric property in tensor manipulation

I have been trying to express $\eta^{\mu\nu}$ in terms of $\eta_{\mu\nu}$ and I have stumble upon the following relation: $\eta^{\mu\nu} = \eta^{\mu\alpha}\eta^{\nu\beta}\eta_{\alpha\beta}$ I can ...
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1answer
83 views

Curvature of Light around a Black Hole [duplicate]

I am in a computer graphics class at my university and for my final project, I have chosen to create a program which renders a simple non-rotating black hole and models the curvature of light around ...
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2answers
83 views

What spacelike, timelike and lightlike really mean?

Suppose we have two events $(x_1,y_1,z_1,t_1)$ and $(x_2,y_2,z_2,t_2)$, then we can define $$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$ which is called the spacetime ...
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0answers
31 views

Wave equation given a metric [closed]

Can you explain me how I can obtain a wave equation given a metric? For example, if I have this metric $$g_{\mu\nu}=diag(-e^{2a},e^{2b},e^{2b},e^{2b})$$ where $a=a(t)$ and $b=b(t)$, how can I ...
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3answers
138 views

Determining whether a space is really three or two dimensional? [closed]

A space purports to be three dimensional with the metric $$dl^2=dx^2+dy^2+dz^2-\left(\frac{3}{13}dx+\frac{4}{13}dy+\frac{12}{13}dz\right)^2$$ How can I show that it actually represents a two ...
4
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0answers
43 views

How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
1
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1answer
81 views

How can I use Einstein's field equations? [duplicate]

Every time I try to find the answer to this question I get redirected to different pages that ultimately do not end up answering my question. I have some understanding of Riemannian geometry but have ...
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1answer
62 views

Does anyone recognize the line element $ds^2 = ( 1 - \frac{2m}{r} )dt^2 + 2 dt dr$?

I've stumbled upon the line element $ds^2 = ( 1 - \frac{2m}{r} )dt^2 + 2 dt dr$. Obviously the corresponding metric tensor has components: $\begin{bmatrix} g_{tt} & g_{tr} \\ g_{rt} & g_{rr} ...
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3answers
144 views

Why doesn't $ds^2 = 0$ imply two distinct points $p$ and $p'$ on a manifold are the same point?

Let's suppose I have a spacetime manifold $M$. Let $p$ be a point on my manifold. Now I move from $p$ to some other point $p'$. Presumably I should have moved some "distance" right? How can I speak of ...
2
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1answer
73 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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0answers
20 views

Decomposition into symmetric and antisymmetric form [closed]

(a) Given a second-rank tensor Tμν, often viewed as an $N \times N$ matrix (for a space of dimension $N$), show by explicit construction that one can always decompose $T_{\mu\nu}$ into a symmetric ...
3
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1answer
95 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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3answers
175 views

Finding the appropriate coordinate transformation given two metrics

Given the two-dimensional metric $$ds^2=-r^2dt^2+dr^2$$ How can I find a coordinate transformation such that this metric reduces to the two-dimensional Minkowski metric? I know that ...
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2answers
75 views

What are $\mu$ and $\nu$ in $g_{\mu\nu}$ metric?

What are $\mu$ and $\nu$ in $g_{\mu\nu}$ metric? Consider the metric $g_{\mu\nu} = \begin{pmatrix} 1 & 0 &0 \\ 0 & r^2 & 0\\ 0 & 0 & r^2\sin^2\theta \end{pmatrix}$
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3answers
444 views

Why is the scalar product of four-velocity with itself -1?

My GR book Hartle says the scalar product of four-velocity with itself $-1$? Consider the definition of four velocity $\mathbf{u} = \frac{dx^{\alpha}}{d\tau}$. Suppose I take the scalar product of ...
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1answer
51 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 ...
4
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1answer
58 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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4answers
175 views

What is Minkowski spacetime?

I was browsing through an article on spacetime when I caught the words Minkowski Spacetime. A Wikipedia search brought me an article too complex for me to totally understand. So what is Minkowski ...
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0answers
39 views

Boyer–Lindquist coordinates

In the Kerr solution to the vacuum Einstein Equation written in Boyer–Lindquist coordinates. Because it is not spherical polar coordinates, $r$ ranges from 0 to infinity does not cover all the space, ...
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1answer
40 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?
3
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1answer
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Inverse Metric Tensor

First the setup: Let $\mathcal M$ be a $2$-dimensional manifold. Let $U_P$ be some open neighbourhood of a point $P \in \mathcal M$. Let $\mathcal F : U_P \rightarrow \mathbb R \times \mathbb R$ be ...
4
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1answer
65 views

Fastest way to find the curvature terms from a given metric [closed]

I want to find the spherically symmetric, static solutions to Einstein's equations $$ R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} = 0 $$ in four dimensions using the metric $$ g_{\mu \nu}dx^{\mu}dx^{\nu} ...
0
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1answer
36 views

Given any metric, how to find the straight line path between two points? [closed]

Say we are given a two-dimensional metric $$ds^2=f_1(x)dx^2+f_2(x)dy^2,$$ for any kind of function. How do we calculate the distance along a straight line path (not the shortest possibly) between, ...
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Why is spacetime not Riemannian?

I apologize if this is a naïve question. I'm a mathematician with, essentially, no upper-level physics knowledge. From the little I've read, it seems that spacetime is Lorentzian. Unfortunately, the ...
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1answer
66 views

Four-vectors and metric tensor

I think it's safe to say that if $x^\mu=(x^0,x^1,x^2,x^3)$, then $x_\mu=(x^0,-x^1,-x^2,-x^3)$. But I don't really understand why one follows from the other. Could someone explain? Also, I've been ...
5
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1answer
81 views

Eddington-Finkelstein coordinates: Why $\ln(r-2m)$ instead of $\ln|r-2m|$?

If one considers the Schwarzschild metric $$ \text d s^2 = -V(r)\text d t^2 + \frac{1}{V(r)}\text d r^2 + r^2 \text d \Omega^2\;,\qquad V(r) = 1-\frac{2m}{r}\;, $$ and introduces the ...
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2answers
50 views

Hermitian Metric and Geodesics

Why isn't general relativity developed with a Hermitian metric and a theory of complex valued paths and geodesics? The concept of arc length and geodesic suffers under a pseudo-Riemannian metric. My ...
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0answers
37 views

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 ...
0
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1answer
43 views

Schwarzschild Solution Convention

In looking at the components of the Schwarzschild Metric, one finds $ g_{00} = (1 - \frac{r_s}{r})c^2 $. Wikipedia states that $r$ is measured as the circumference, divided by $2π$, of a sphere ...
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4answers
553 views

Can a metric in General Relativity, Supergravity, String Theory, etc., be asymmetric?

Why is it that all problems I encountered until now have metrics that when represented in a matrix form turn out to be symmetric? Aren't there asymmetric matrices representing some metrics?
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5answers
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Is $ds^2$ just a number or is it actually a quantity squared?

I originally thought $ds^2$ was the square of some number we call the spacetime interval. I thought this because Taylor and Wheeler treat it like the square of a quantity in their book Spacetime ...
1
vote
1answer
85 views

What is the covariant basis around a Schwarzschild black hole?

First of all, I'm not interested in time for this question. So lets consider a 3-manifold whose metric is the spatial part of the Schwarzschild metric, so it has the event horizon and the singularity ...
2
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2answers
169 views

Why do things slow down when you move faster, rather than speed up?

I've been trying to get to grips with SpaceTime. As I understand it, we move at a set rate through spacetime. Any increase in our rate of travel through space results in a decrease in our rate of ...
5
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2answers
149 views

Metric tensor in SRT

I just read on this webpage that we have (click me) $g_{\alpha \beta} = g_{\alpha}^{\beta} = g^{\alpha \beta}.$ Now, although I understand that the first and the last one are equal, I don't think ...
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0answers
26 views

Norm of Killing vector field

Let us suppose we have a Killing vector field with $X^a = 1/2$ and $X^b = 1/3$ and $g_{ab}=1$ where the other $c$ and $d$ components are zero. Now we want to find its norm: The formula for finding ...
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1answer
65 views

Geodesic deviation

In S. Carroll Lecture Notes on General Relativity, chapter 6, pages 152-153 we have equation (6.62) $$\tag{6.62} \frac{\partial^2}{\partial t^2} S^\mu=\frac{1}{2} S^\sigma \frac{\partial^2}{\partial ...
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1answer
84 views

How to find a metric of a general observer?

Yes, that's it. How to find a particular metric of an observer in general relativity? Let's say we have a static metric: ...
3
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1answer
80 views

Naturalness of tensor fields in general relativity?

In the third chapter of the book The Large Scale Structure of Space-Time, the authors say regarding the matter fields in general relativity: These fields will obey equations which can be expressed ...
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1answer
35 views

isotropy of 3-space and spacetime metric

The most general spacetime metric is given by $$ds^2=g_{\mu\nu}dx^\mu dx^\nu=c^2dt^2+g_{0i}dtdx^i-g_{ij}dx^i dx^j$$ Why does the second term said to violate isotropy of 3-space? It is true that, ...
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1answer
74 views

Why does $\frac{d\tau}{d\sigma} = L$?

I am given a (3+1)-dimensional spacetime that has the line element \begin{equation} ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1} dr^2 + r^2 d\phi^2 \end{equation} ...