Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [metric-tensor]

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.

0
votes
1answer
66 views

Toy almost everywhere flat metric

I was looking into this paper from 2003 by Krasnikov - “The quantum inequalities do not forbid spacetime shortcuts” - and apart from the fact that I find it very scetchy without many details and hard ...
3
votes
1answer
54 views

Metric for a Collapsing Disk and FLRW

I have to obtain a time-dependent metric for a disk which satisfies a simple differential equation but I am stuck with making sure that the physics is correct. To be explicit, I'll first describe a ...
0
votes
0answers
48 views

Infinitesimal squared in metrics [migrated]

Metrics are often formulated by appealing to the square of an infinitesimal quantities. Examples of such are: $$ (ds)^2=(dx)^2+(dy)^2 $$ or $$ ds^2=dx^2+dy^2 $$ or $$ d(s^2)=d(x^2)+d(y^2) $$ ...
0
votes
1answer
50 views

Solving differential equation for the Schwarzschild metric with cosmological constant

How do we solve for Einstein's equation in the vacuum with a cosmological constant, in the static spherically symmetric situation? Attempt: Following Sean Carroll's Spacetime and Geometry (p.195), I ...
-1
votes
0answers
49 views

A vector that is both parallel and perpendicular to another vector

The following is an excerpt from "Einstein Manifolds" by Besse Suppose we have a freely falling test particle, described by a timelike geodesic $\gamma$. A nearby (infinitesimally close) freely ...
1
vote
3answers
68 views

Difference between pseudo-Riemannian metric and the solution of Einstein field equations

Nakahara in his book on the Geometry and Topology introduces the pseudo-Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication ...
2
votes
1answer
42 views

Determinant of ADM metric

I am studying inflation and for the calculation of the bispectrum we are using the ADM formalism where the metric is the following form: $$g_{\mu\nu}=\begin{bmatrix}-N^2+N^iN_i&N_i\\N_i&h_{ij}...
1
vote
1answer
87 views

When to use an orthonormal basis in GR?

I am working on a problem in the textbook Gravity: An Introduction to Einstein's General Relativity by Hartle. The problem is to show that an observer moving radially through the throat of the ...
0
votes
1answer
44 views

Four momentum squared and collisions

So, I am not asking is the square of four-momentum of a particle an invariant to Lorentz trasnformations, but rather,is it invariant in dynamic situations? It seems to me that this also has to hold. ...
1
vote
0answers
24 views

Question about Lie derivative of connection [closed]

This is from Hughston and Tod, Ex 8.5. Please give me an idea how to start with this. If $\mathcal{L}_V g_{ij} = 2\phi g_{ij}$ then prove that $$ \mathcal{L}_V\Gamma^{i}_{jk} = \phi_{,j}\delta^{...
2
votes
1answer
53 views

Is there only one vacuum solution of the Einstein equations?

I am thinking about this: A vacuum solution means vanishing Ricci tensor. The Ricci tensor is a contraction of the Riemann, which itself involves contains second derivatives of the metric. Thus they ...
0
votes
1answer
38 views

Is Levi cita tensor an invariant in curved space?

The Minkowski metric and Levi cita tensor is an invariant quantity in Euclidean flat space. But in curved space metric tensor varies. Analogous to it, is the Levi cita tensor varies in any Non-...
1
vote
1answer
85 views

Coordinate transformations in general relativity

Let's assume a non-rotating point mass with mass $M$. A non-massive object travels with constant velocity $\mathbf{v}_t$, with respect to the point mass, in the vicinity of the point mass. A non-...
0
votes
0answers
72 views

Solving the biharmonic equation $\square^2f(\vec{x}) =0$ on a curved space

How to solve the biharmonic equation $\square^2f(\vec{x}) =0$ on a curved space (e.g. $d$ dimensional sphere or hyperboloid)? I was thinking in the following line : I know how to solve $\square f(\...
3
votes
3answers
64 views

Unique null geodesic between two points

Given two points in Lorentzian spacetime $p,q\in M$, is it true that there is only a unique null geodesic (up to affine reparametrization) that connects that the two points? On the one hand, it seems ...
2
votes
1answer
71 views

Help with the river model of black holes

I've been reading the paper "The river model of black holes" by Andrew Hamilton et. al., but I've been unable to derive the paper's results on the tetrad frame connection coefficients. I must be doing ...
2
votes
1answer
59 views

What's the variation of the Christoffel symbols with respect to the metric?

By the Leibniz rule, I expected it to be $$\delta \Gamma^\sigma_{\mu\nu} = \frac 12 (\delta g)^{\sigma\lambda}(g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu}-g_{\mu\nu,\lambda}) + \frac 12 g^{\sigma\lambda}(\...
1
vote
0answers
76 views

Perturbation to the flat space metric

from the geodesic equation for non-relativistic case where $$v_i\ll c$$ $$\frac{dx^i}{dt}\ll1,{\rm for }\ c =1$$ $$\frac{dx^i}{d\tau}\ll\frac{dt}{d\tau}$$using this the geodesic equation for proper ...
6
votes
4answers
85 views

When is the value of spacetime interval $ds$ negative?

The spacetime interval in special relativity, $ds$, is defined as $$ ds^2=c^2dt^2-dx^2-dy^2-dz^2 $$ with the $(+,-,-,-)$ Minkowski sign convention. The value of $ds^2$ can be positive, zero, or ...
1
vote
2answers
58 views

Transform mixed vielbein expressions to $\sqrt{-g}$ times traces in massive gravity?

In the Massive Gravity review by Claudia de Rham the massive gravity action is given by with mass potential in vielbein formulation. Equivalently, the same action can then be described by with I ...
0
votes
0answers
44 views

Question about the geometric structure of Newtonian mechanics

My point here is about the mathematical structure of Classical pre-relativistic physics and general relativity (GR). It became more clearly, after GR, about the fact that pseudo-riemannian are a nice ...
2
votes
1answer
101 views

How to calculate initial conditions to integrate a null geodesic

Suppose, this is the line element of a FLRW metric, $$ ds^2 = -[1 + 2ψ(t,x_i)]dt^2 + a^2(t) [1 - 2ϕ(t,x_i)]dx_i^2 $$ and the geodesic equation is, $$ \frac{d^2x^α}{dλ^2} = - Γ_{βγ}^α \frac{dx^β}{dλ} \...
3
votes
2answers
66 views

Spatial and temporal dimensions orthogonality

It seems that the spatial dimensions are orthogonal: a particle can move along one axis without changing its position in relation to other two axes. It seems that the temporal dimension is somewhat ...
-2
votes
1answer
45 views

Time difference between all particles and waves [closed]

Since all elementary particles and waves were created simultaneously in the big-bang (t0) would there be any time difference between any interacting elementary particles and/or waves after t0? I'm ...
1
vote
0answers
58 views

Proper time and its definition [duplicate]

Why is proper time defined to be of opposite sign of the space time interval ds? Shouldnt it be exactly the space time interval? My question was not about time having opposite sign it was about ...
0
votes
1answer
46 views

How does the Equivalence Principle imply that derivatives of the metric vanish in a freely falling frame?

Why do the first derivatives of $g_{\mu\nu}$ vanish in a freely falling coordinate system? I would like to start from the Equivalence Principle that for any point in spacetime there exists a locally ...
0
votes
1answer
45 views

What are some of the reasons for raising/lowering indices of a tensor?

In Dirac's paper: Classical theory of radiating electrons, he decides to raise and lower the indices on the same object multiple times: \begin{align*} \frac{\partial{A_{\mu}}}{\partial{x_{\mu}}} &...
1
vote
0answers
61 views

Gauge invariance in GR perturbation theory

I have been following this video lecture on how to find gauge invariance when studying the perturbation of the metric. Something is unclear when we try to find fake vs. real perturbation of the ...
1
vote
1answer
64 views

Finding out “the” coordinate transformation to Minkowski metric

This is a homework problem. I am given the following metric: $ds^2 = Fc^2dt^2 - \frac{1}{F}dr^2 - r^2d\phi^2 - dz^2~;~F>1, F = F(r)$ They ask me to find "the" coordinate transformation that will ...
2
votes
2answers
107 views

Relativity in 2+1 or 4+1 dimensions

Whereas we have experience of relativity working with 3 spacial dimensions and one of time would there be similar rules affecting a two dimensional space and one of time or even with a 4 space and 1 ...
0
votes
1answer
28 views

Local mass function in spherically symmetric spacetime

I am studying the paper `Inflation and de Sitter Thermodynamics' at https://arxiv.org/abs/hep-th/0212327 . I have problems with the way they define a local mass function in a general spherically ...
2
votes
1answer
61 views

What is time-like path in GR?

My understanding: Given a metric, at each point of spacetime, there is a tangent vector u that maximize the quantity $g_{ab}u^au^b$, which is the proper-time length. Does it mean at each point, there ...
1
vote
3answers
63 views

How to get space component of weak field (linearized) metric?

For Minkowski space with a weak gravitational field the metric takes the form $$ ds^2 = (1+2\phi)dt^2 -(1-2\phi)(dx^2+dy^2+dz^2), $$ where $\phi$ is the Newtonian gravitational potential. You can get ...
3
votes
0answers
157 views

Metric of a cross-section in General Relativity [migrated]

Consider a finite closed region $V=(x,y,z)$ as a simply-connected subset of a 3-dimensional flat Euclidean space ${\Bbb R}^3$ with the metric $\text{d}s^2=\text{d}x^2+\text{d}y^2+\text{d}z^2$. A ...
0
votes
0answers
39 views

Determinant of induced metric

I've got the following set up for the induced metric: How can I see that the determinant of such induced metric is the one from the following expression?
0
votes
0answers
24 views

Possible maximally symmetric 3D spaces

I was watching Neil Turok's lectures on General Relativity. After introducing the Einstein equation, he tries cosmology and postulates "The space is assumed to be isometric and homogenous." Then he ...
1
vote
1answer
93 views

Does the vierbein contain any extra information?

The vierbein from General relativity has $D(D+1)/2$ independent components when accounting for the $O(3,1)$ gauge symmetry. The metric has the same degrees of freedom. But does the vierbein contain ...
0
votes
0answers
22 views

Covariant divergence in GR [duplicate]

In Carroll's Spacetime and Geometry (§3.2) I found $$\nabla_\mu V^\mu =\partial_\mu V^\mu + \Gamma^\mu_{\mu\sigma}V^\sigma =\frac{1}{\sqrt{\left|g\right|}}\partial_\mu \left(\sqrt{\left|g\right|}V^\mu\...
0
votes
2answers
89 views

How does 4-vector notation work?

In particle physics we are going over 4-vector notation. However, my background on this is a little shaky, and I'm having difficulty differentiating the notation and visualizing what it actually means....
1
vote
2answers
45 views

Constraint on vierbein vectors

Is it reasonable to choose the vierbein frame $e_{a}^{\mu}$, with the following constraint being imposed: $e_{a\mu}^{\quad;\mu} = 0$? If yes, how one can find such vierbein vectors for the Kerr metric....
0
votes
1answer
60 views

Deriving gravitational acceleration from the Schwarzschild metric

I am supposed to derive the newtonian equation for gravitational acceleration ($-GM/r^2$) from the Schwarzschild metric $$ds=-(1-\frac{r_s}{r})c^2dt+(1+\frac{r_s}{r})^{-1}dr+0$$ where $$r_s=\frac{2GM}{...
3
votes
1answer
54 views

Accelerated frame approximation in Schwarzschild metric far from the horizon

It is clear to me that if I take the Schwarzschild metric $$ds^2 = \left(1-\frac{2M}{r}\right)dt^2 - \left(1-\frac{2M}{r}\right)^{-1} dr^2$$ and choose $\rho = 2\sqrt{\frac{r}{2M} -1}$ then I get the ...
1
vote
3answers
116 views

Derivation of Inverse Lorentz Transformation in Index Notation

To review my special relativity I tried to work out the inverse lorentz transformation explicitly. This led to a lot of confusion; I would like to ask what the issue was with the assumptions I made in ...
0
votes
2answers
86 views

Are Euclidean solutions also solutions of Einstein's equations?

In Special relativity we have the metric $(+---)$. But in General relativity we have a metric tensor $g$. In the equations themselves there doesn't appear to be anything that tells you what the ...
4
votes
1answer
59 views

The sphere $S^d$ is Euclidean space $E^d$ with infinity identified as a single point

I'm reading about anti de Sitter spacetime, and I found the following statement: $$ds^2 = \frac{1}{\cos^2 \psi} \big( -dt^2 + d\psi^2+ \sin^2 \psi d\Omega_{d-2}^2 \big).$$ Thus, the spatial ...
0
votes
0answers
43 views

deSitter spacetime metric and curvature

I have to compute the metric of an hyperboloid given by $-(X^0)^2+(X^1)^2+(X^2)^2+(X^3)^2=H^{-2}$ in 5D Minkowski spacetime using the following coordinates: $$X^0=H^{-1}\sinh(Ht)\sqrt{1-H^2r^2}$$ $$X^...
1
vote
1answer
70 views

Index position when varying an action with respect to the metric

I'm confused about where we should put tensor indices when we vary an action wrt the metric. For example, if I have in the Lagrangian a term such as $$ A_{\mu\nu}B^{\mu\nu}, $$ do I necessarily have ...
0
votes
0answers
42 views

Proving two space-time intervals are equivalent with matrix algebra

η=$ \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $ v=$ \begin{bmatrix} ct\\ x\\ y\\ z \end{...
0
votes
2answers
70 views

Spacelike and timelike intervals confusion

I'm confused about this, specifically the spacetime interval. A timelike interval is one in which 2 events can be related to each other in a given reference frame within its light cone, that is, it ...
0
votes
1answer
75 views

How is four-velocity automatically normalized?

This is a page from Sean Carroll's Spacetime and Geometry.There is a line in this page which says that the four velocity is automatically normalized.This absolute normalization is a reflection of the ...