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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.

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45 views

Diffeomorphism invariance in special relativity

Suppose space time is the manifold $M $ isomorphic $ \mathbb{R^4}$ whit the metric $-\eta_{00}=\eta_{11}=\eta_{22}=\eta_{33}=1$ in the Cartesian coordinates $\Psi(p)=(x^0,x^1,x^2,x^3)$ for $p \in M $ ....
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Point particle and angular deficit

I would like to understand in what sense an angular deficit can be interpreted as a point particle. Typically, if you have a metric in polar coordinates such as: $$ds^2 = -dt^2 + a^2(t,r) dr^2 + r^2 d\...
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Metric tensor $g$ for static gravitational field referred to static coordinate system

Assume there is static gravitational field. I want to deduce that there exists a coordinate system where $$g_{m0}=0, \quad m=1,2,3.$$ Is this a reasonable result? Why would it be contradictory if ...
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Differentiation of metric tensor in new coordinate

I want to understand the explicit meaning of $g_{\mu'\nu',\lambda}=0$ where unprimed coordinates are coordinates of the the original coordinate systems and primed ones are for new coordinate system. ...
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2answers
51 views

Differentiation of the determinant $g$

Let $g$ be the determinant of the metric tensor. I want to derive the following equation $g_{,\nu}=gg^{\lambda \mu}g_{\lambda \mu,\nu}$. It is said that $gg^{\lambda \mu}$ is a cofactor, but I can't ...
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83 views

The definition of quantities in special relativity as upper-index or lower-index

My question is for Minkowski metric $\eta_{\alpha\beta}=\mathrm{diag}(1,-1,-1,-1)$. While defining quantities like the four potential, four momentum or even space-time interval for that matter, why do ...
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Is the relativistic time dilation compatible with the gravitational field of a moving massive planet?

Assume that a satellite orbits a massive planet/star at a distance $r^\prime$ away from its center in a circular path in plane $x^\prime y^\prime$. Suppose that the gravitational field is $g^\prime$ ...
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1answer
91 views

Motivation for tensor theory of gravity

In class we were shown that $$\rho = \frac{dm}{dV}$$ has the transformation properties of the 00 component of a rank 2 tensor. So we'd like to turn the classical Poisson equation for gravity into a ...
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1answer
29 views

Isometry of Riemann sphere?

The complex metric on the Riemann sphere is given in the Wikipedia article to be $$ds^2=\frac{4}{(1+\zeta\bar \zeta)^2}d\zeta d\bar \zeta$$ while the sphere should be mapped to itself under $SL(2,\...
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Trajectories in space

I want to say that a set $T$ of vectors in $R^{\,4}$ is a "trajectory" if there is an interval $I$, and continuous functions $x,y,z$ on $I$, such that $T$ is the set of $[t,x(t),y(t),z(t)]$ for $t$ in ...
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62 views

Geometrical interpretation of curvature invariants

Consider a Riemannian manifold. It is possible to describe it by curvature invariants. Now, is there any geometrical description (intuition) for simple invariants such as scalar curvature, Ricci ...
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87 views

Can a “time dimension” be part of a spherical topology?

I've heard it speculated that the spatial dimensions of the universe is a 3-sphere. Or a 3-torus. But usually, I guess, it's assumed that the "time" dimension just has its own geometry, like a line, ...
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26 views

How to find the mixed tensor, contravariant tensor and tensor trace of $F$

I have a question in particle physics that ask me to find the mixed tensor, contravariant tensor and tensor trace of $F$: Our professor didn't teach us that much about the math of tensor, which makes ...
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2answers
82 views

Direct derivation of point-like particle metric in GR

The usual way to derive metric of a point mass in general relativity is (to my knowledge) based on assuming specific form of the metric that reflects spherical symmetry and independence on "time" (...
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1answer
50 views

Geodesic Equation from Coordinate Transformation

Let $\xi^a$ be the usual coordinates and $x^\mu$ the new coordinates, both flat. Now we know that since the metric is flat, $$ \frac{d^2\xi^a}{d\tau^2} = 0 $$ $$ \Rightarrow \ \frac{\partial}{\...
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57 views

How to find a normal to an hypersurface

I have to apply the Israel junction conditions in a region in which a hypersurface with O(3) symmetry separates two spacetime with Schwarzschild metric (with masses $M_+$, the exterior one, and $M_-$, ...
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41 views

Coordinate-free representation of invariant interval [closed]

Suppose event B is at the origin of space-time diagram. Let A and C be arbitrary events (A is in the backward light cone and C is on the forward light cone with B at the origin). The world line ...
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How to get this metric tensor perturbation equation? (Gravitational wave)

(I need some time to come back to re-edit this post) LHS of equation 7.37 is gauge invariant The Twisted H symbol is comoving hubble parameter, the h_ij is metric perturbation, I do not know how to ...
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1answer
53 views

Tensor analysis: confusion about notation, and contra/co-variance

I'm learning about tensors in the context of special relativity, and I'm a bit confused some notation. I understand a four-vector is a four dimensional vector, which is written in the form $(ct, x, y,...
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How can I show that the inverse of the induced metric $h_{\alpha \beta}$ is $h^{\alpha \beta}$?

So I was reading through Becker, Becker, Schwarz and there is a line in the second chapter that states that $h^{\alpha \beta} = (h_{\alpha \beta})^{-1}$ where $h_{\alpha \beta}$ is defined as: $$h_{\...
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1answer
79 views

Different versions of the Robertson-Walker Metric

One form of the Robertson-Walker metric is $$ds^2 = c^2dt^2 - a(t)^2[d\chi^2+ S_k(\chi)^2(d\theta^2 + \sin^2\theta ~d\phi^2)]\tag{1}$$ $$\\$$ Considering curvature, where k = 0 , +1, -1 for flat, ...
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Basic question about units of velocity and speed of a curve on a smooth manifold

Frederic Schuller says that velocity has units in Hertz in The WE-Heraeus International Winter School on Gravity and Light. He says: \begin{align} [v^a]&=\frac 1 T \\ [g_{ab}]&=L^2 \\ \Big[\...
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2answers
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How can I think of the flat space metric tensor as a multilinear function?

I'm pretty new to the idea of tensors, and I'm having a bit of confusion with how to think about the flat space metric tensor in special relativity. I understand that a good way to think about ...
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1answer
35 views

Finding the inverse metric of a metric close to Minkowski metric [closed]

Let $g_{uv}=\eta_{uv}+h_{uv}$ be a metric with $\mid h_{uv} \mid$ very small so that the metric is close to the Minkowski metric. Then we can write the inverse metric $g^{uv}=\eta^{uv}+k^{uv}$ with $\...
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80 views

Pertubation of Riemann tensor in a general curved space-time

It is a direct and simple question. I am fully developing the perturbation of Einstein Field Equations, and I need to calculate the perturbation of the Riemann tensor. However the background metric is ...
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Imaginary term solution in the limit $v\ll c$ of relativistic Lagrangian

The relativistic action is $$ S=- m \int_a^b d s. $$ With metric $ds^2=dx^2 - dt^2$, we get: $$ \begin{align} S&=\pm m \int_a^b \sqrt{dx^2-dt^2}\\ &=\pm mc\int_a^b dt\sqrt{\left(\frac{dx}{...
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Normal to an hypersurface and junction conditions

I'm looking for applying the Israel junction conditions to the collapse of a spherical region filled with pressure-free dust. The metric inside is a k=1 Friedmann model $$ds^2_{-}=−d\tau^2 +a^2(\tau)�...
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43 views

Confusion about stress-energy tensor in 2D gravity when reformulated as a CFT

I'm trying to verify that I obtain the same stress-energy tensor for a simple 2D gravity theory and the associated CFT reformulation. However, the results are not agreeing which leads me to believe I'...
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1answer
41 views

Uniqueness of affine connections

This is a problem from Carmelli book on general relativity. the conceptual problem is, given a spacetime, and hence a metric, can there exist more than one affine connection for which one can take ...
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Differentiability of a Metric Tensor

As an introduction to Metric Tensors, I read the conditions to be met for a metric includes it being differentiability class $C^2$ i.e. all second-order partial derivatives of $g_{ij}$ exists and are ...
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Is the “spacetime” the same thing as the mathematical 4th dimension?

Is the "spacetime" the same thing as the mathematical 4th dimension? We often say that time is the fourth dimension, but I am wondering if it's means that time is like the fourth geometrical axis, or ...
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2answers
113 views

Why is it necessary that different observers agree on the value of the spacetime interval $ds^2$?

What's the physical reason that all (inertial) observers agree on the value of the spacetime interval $$ds^2 = (c dt)^2 - dx^2 - dy^2 -dz^2 \, ?$$ What would be the physical implications if different ...
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53 views

How come this is a gravitational plane wave?

In Wikipedia's Gravitational plane wave article a metric for a gravitational plane wave is given by \begin{equation} ds^{2}=\left[a\left(u\right)\left(x^{2}-y^{2}\right)+2b\left(u\right)xy\right]du^{2}...
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2answers
78 views

Confusion re: geodesics, connections, and straightness

In my readings in GR I often come across geodesics characterized as "straightest possible curves." This characterization confuses me. I'd like some clarification as to whether I'm understanding the ...
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1answer
49 views

The description of a universe with one or two objects

I understand that in general relativity, we describe the universe by the relationship between objects "in it". I quote "in it" because in this model, there is no background where the objects exist. ...
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1answer
35 views

Equation in Current vector in a Klein Gordon Equation

I'm trying to get the current vector $J^\mu$ of a Klein-Gordon equation: $$\Psi^* \Box \Psi =\Psi^* \partial^{\mu} \partial_\mu \Psi= \partial^{\mu}(\Psi^*\partial_\mu \Psi)-\partial^\mu \Psi^*\...
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When exactly is a dimension spatial?

I every so often hear claims like: M-Theory predicts that there are 10 spatial dimensions! Now I'm not really sure what these claims mean. There are three spatial dimensions that I normally ...
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0answers
41 views

Derivation of Isometries of $AdS_3$ in Poincare Coordinates

We know that $SO(d,2)$ is the isometry group of $AdS_{d+1}$. Let's only consider $AdS_3$ in this question. In Poincare coordinates ($r,t,x)$, these can be grouped as follows : Two translations $$r'=...
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1answer
57 views

Is it possible to make a metric describing a non-stationary system?

I know from the Schwarzschild metric that it is a stationary solution to einstein`s equations described by its metric. But what if I have a spacetime that evolves with time? Can I build a metric to ...
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1answer
68 views

Electromagnetic field tensor (and other tensors) with different sign conventions

In Wikipedia the components of the EM Field Tensor are listed as $$F^{\mu\nu}=\left( \begin{array}{cccc} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z &...
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1answer
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Is Schwarzschild's solution in his original paper consistent with current solutions? [duplicate]

I was reading the Schwarzschild's original paper where he derives the Schwarzschild metric for the first time(The english translated version found in arXiv : On the Gravitational Field of a Mass ...
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1answer
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Why does a wormhole have this metric?

I asked this (or something similar) within another question and was asked to post it separately, so here goes: The metric for flat Minkowski space is: $$ds^2 = -dt^2 +dr^2 +r^2(d\theta^2+d\phi^2\sin^...
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1answer
80 views

Flat spacetime with curved metric? Small change in metric drastically changes geometry? GR

I'm about to be taking an undergrad course in GR. I'm trying to get ahead of the game, and I've hit a problem. The metric for flat Minkowski space (using the $-+++$ signature and $c=1$) is: $$ds^2 = ...
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1answer
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Pseudo-Riemannian 2D manifold (visualize time curvature)

My goal is to visualize somehow the curvature of time, as opposed to the curvature of space. I know that we generally talk about spacetime curvature altogether; however, the fact that spacetime has ...
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51 views

Example of a metric of the form $ds^2 = g_{vv}(v,x)dv^2+2g_{vx}(v,x)dv\,dx+r^2(v,x)d\Omega^2$ with trapped region

Is there a known example of a spherically symmetric metric, especially one describing a spacetime containing trapped region/black hole, which takes the form $$ds^2 = g_{vv}(v,x)dv^2+2g_{vx}(v,x)dv\,dx+...
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1answer
38 views

Gauge fixing of Polyakov Action

In the Gauge fixing of Polyakov action we do general coordinate transformation where we take the transformation stated below $$h_{\alpha\beta} = e^{\phi(\sigma)}\eta_{\alpha\beta}.$$ But here the ...
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1answer
85 views

Stuck Solving MTW Gravitation Problem 20.5

I am stuck on exercise 20.5 part a) from Misner, Thorne, and Wheeler's Gravitation chapter 20. The Einstein summation convention is used throughout this post. Problem Statement Calculate $t^{\...
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1answer
268 views

Is a Wick rotation a change of coordinates?

My understanding is that a Wick rotation is a change of coordinates from $(t,x) \rightarrow (\tau , x)$ where $\tau = i t$. In the $(t,x)$ coordinate system, the Minkowski metric has components $ \...
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1answer
70 views

Geodesic equation and spatial variation of time

I am trying understand the interpretation of geodesic equations. For simplicity, let us take a metric $$ds^2 = g_{00}(x)dt^2 + a(x,y,z)(dx^2 + dy^2 + dz^2).$$ I interpret the metric to be a spacetime,...
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1answer
35 views

Raising and lowering the indices of a perturbed metric

I am looking at a metric which is defined as (Eq 2.4 Glampedakis & Babak) $$ g_{\mu \nu} = g_{\mu \nu}^K + \epsilon h_{\mu \nu}$$ where $g_{\mu \nu}^K$ is the original unperturbed metric (Kerr) ...