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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How do we make sense of $F^{\mu\nu}F_{\mu\nu}$? The book just assumes I understand it

Why are these upper and lower indices and what does that mean. I can't interpret the term with upper indices.
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Understanding Killing vectors of FLRW metric

I am trying to understand how to calculate the Killing vectors of FLRW metric \begin{equation} ds^2 = dt^2 - R(t)^2\left( \frac{dr^2}{1 - k r^2} + r^2 d\theta^2 + r^2 \sin\theta d\phi^2\right). \end{...
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Notation for the metric of $\rm dS_4$ and/or $\rm AdS_4$

4D de Sitter and anti-de Sitter spaces may have their metrics inferred from the induced metric on a hyperboloid embedded in 5D Minkowski space: $$ -( x^0)^2+( x^1)^2+( x^2)^2+( x^3)^2+( x^4)^2=\pm \...
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1 answer
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Contravariance and covariance of vectors

My main source of confusion is the following. Suppose I have a scalar potential $V(x,y,z)$. The electrostatic field for this potential is $ -\vec{E} =\vec{\nabla}V = \frac{\partial{V}}{\partial{x}}\...
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How (if) can we connect a 2D "throat" piece of a wormhole to two hyperbolic 2D manifolds?

This question wad closed on the mathematics site, as it lacked clarity. So I try my luck here. My question is cosmology-inspired. Imagine two 2D hyperbolic manifolds. I connect them by a manifold like ...
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How (if) is the metric of the quantum vacuum different from the metric of the classical vacuum?

The classical vacuum, with no matter or energy in it, has a flat metric. Meanwhile we know that the classical vacuum is a chimera. There are lots of things going on, eventhough its called virtual. ...
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Where is the Lorentz signature enforced in general relativity?

I'm trying to understand general relativity. Where in the field equations is it enforced that the metric will take on the (+---) form in some basis at each point? Some thoughts I've had: It's baked ...
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What's the time component of the metric of a saddle-shaped space? [closed]

Two masses in a negatively curved spacetime, starting at rest, accelerate away from each other. There is no equivalent elevator experiment to simulate this kind of acceleration (or is there?). In the ...
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Are there types of spacetime that have no symmetries?

We derive the most basic laws of physics from several fundamental symmetries (those from Noether's theorems, gauge symmetries, Lorentz symmetry...). But are there any types of spacetime where no ...
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Transform differential equation in other spaces with different metrics

At first I'm sorry for this poor question but I'm pretty new to this area of mathematics. Lets say I have an differential equation like $\frac{\partial f(x,t)}{\partial x} = const.$ in a 2 dimensional ...
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Could time be a secondary effect due to curvature of space?

In general relativity, four-dimensional spacetime is considered and curvature is calculated for spacetime, not only space alone. However, looking deeper into the equations, many sources of symmetry ...
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Could 2D spacetime be seen as an embedded manifold?

In a chart of a 2-dimensional spacetime manifold, we can write the rule giving separation between two points as: $$ ds^2 = - dt^2 + dx^2.$$ Could we use this to imagine how space time looks like a ...
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How to write down the metric explicitly?

I know that we can derive Schwarchild metric by imposing torsionless condition. if the torsion vanish, we can write down spin connection explicitly by veilbein. After variation to veilbein of Einstein-...
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Normal to the hypersurface

I have a given metric and I want to choose a spacelike hypersurface and find the normal to that hypersurface. I know that if the hypersurface is spacelike, then the normal is timelike. The given ...
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How do we figure out what is the right geometry of space?

In page-319 of Visual Differential Geometry, the following is written: When we speak of a solution to Einstein's equation, we mean a geometry of space time (defined by it's metric) that satisfies the ...
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How to calculate Proper Distance as an arc length in Schwarzschild metric?

I am trying to determine the method to calculate proper distance with constant time and radius in Schwarzschild Geometry. With only $\theta$ and $\phi$ being variable. I think it involves integrating ...
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The Derivation of the Schwarzschild Solution

I went to this site to find the solution. However, I have a few questions about where these equations come from. In the category of Assumptions and Notations, what equation gives you $\partial_tg_{\...
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1 answer
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How to simplify the process of calculating spacetime geodesics?

I want to study the movement of a particle along geodesics in an expanding universe with metric (FRW metric) $$ ds^2 = -dt^2 + a^2(t) \left( \dfrac{1}{1-kr}dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\...
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Linear Momentum in General Relativity

My question is, does a particle moving in a straight line at constant velocity through empty space create "frame dragging" that would tend to entrain other bodies in the direction of its ...
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2 answers
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Does the stress energy tensor scale with the metric tensor?

Question I had some thoughts from a previous question of mine. If I have a metric $g^{\mu \nu}$ $$g^{\mu \nu} \to \lambda g^{\mu \nu}$$ Then does it automatically follow for the stress energy tensor $...
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Change of Metric Under Coordinate Transformation

Under a local change of coordinates $x\to x'=x+\delta x$, the metric transforms as $$g_{\mu \nu}^{\prime}\left(x^{\prime}\right)=g_{\lambda \rho}(x) \frac{\partial x^{\lambda}}{\partial x^{\prime \mu}}...
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Riemann curvature tensor in an inertial frame

My understanding is that the mathematical definition of an inertial frame at $x_0$ is a choice of coordinates s.t: $g_{\mu\nu}(x_0) = \eta_{\mu\nu}(x_0)$ $\partial_\rho g_{\mu\nu}(x_0) = 0$ I've ...
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2 answers
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Invariance of spacetime interval by Schutz

I am reading the book on General Relativity by Bernard Schutz. In it he proves the invariance of the interval in special relativity using the following argument. $S^2=0$ for all light-like paths. This ...
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Dimensionless bulk coordinate in AdS?

I just had a look at the AdS-C metric that can be expressed as follows $$\begin{equation} ds^2 = l_4^2 d\sigma^2 + \frac{l_4^2}{l_3^2} \cosh^2(\sigma) \left( -f(r)dt^2 = f(r)^{-1} dr^2 + r^2d\theta^2 \...
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What is the evidence that gravitational fields don't sum up as a superposition?

Einstein's field equations are non-linear. Gravity gravitates (self-interacts). It's very complicated to solve Einstein's field equations for more than one central object. That are keystones in ...
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1 answer
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Trying to derive newtonian potential for Schwarzschild interior metric [closed]

I am using the book "A first course in general relativity" by Bernard Schutz. On page 267 he derives equation 10.54 but leaves out some steps that I am trying to do myself. The following is ...
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Physical significance of metric compatibility

When we try to construct a covariant derivative, we impose several conditions on it so that the resulting derivative is unique. However, I can't make sense of the condition of metric compatibility. I ...
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2 answers
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Understanding EFE: RHS linear, LHS not?

Einstein's field equations are nonlinear. That means it is not allowed to add up the metric tensors. However, on the RHS of the field equations, there is only the stress-energy-momentum tensor, and it ...
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Killing field with a time dependent metric (including $g_{00}$)

Let's suppose that (in cartesian coordinates) $$g_{\mu\nu}=diag(-f(t)^2, g(t)^2, g(t)^2, g(t)^2).$$ So that all of the components of the metric are dependent on coordinate time. If we produce a ...
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Is this version of Einstein field equations linear?

While playing around with the Einstein field equations and trying to derive the Kerr metric, I came across the following derivation from Einstein's field equations: $$R_{\mu\nu} = 8\pi \left(T_{\mu\nu}...
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Finding the proportionality constant in $\varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho}\propto \varepsilon^{\lambda\rho}$

We can show that the contraction of some arbitrary $2\times2$ matrix $A_{\mu}^{\ \lambda}$ with the Levi-Civita symbol is once again antisymmetric \begin{align*} \varepsilon^{\mu\nu}A_\mu^{\ \lambda} ...
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What is a signature of pp-wave metric?

pp-wave spacetime metric is defined in Brinkmann coordinates as $$ds^2 = H(u,x,y)du^2 + 2 du dv + dx^2 + dy^2.$$ Since it's lorentzian (https://en.wikipedia.org/wiki/Pp-wave_spacetime), I wonder what ...
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1 vote
1 answer
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I'm confused about the number of Killing vectors in Schwarzschild metric

I'm trying to perform a calculation to derive the Killing vectors of a spherically symmetric metric (so I use the Schwarzschild metric without loss of generality because the Birkhoff theorem tells me ...
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1 answer
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Metric tensor relations

I need to check some simple tensor relations for continuing my calculations. Please write me if you think anyone is incorrect. P.S.: I know this is very simple question, but i need to be assured about ...
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Different AdS metrics in global coordinates

In David Tong's lecture notes I came across the folloing AdS metric in global coordinates \begin{equation} ds_3^2 = \left( \frac{dr^2}{\frac{r^2}{R^2} + 1 } - \left(\frac{r^2}{R^2} + 1 \right)dt^2 + ...
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1 answer
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Components of the fully contravariant Kronecker Delta in Schwarzschild Metric

I thought the kronecker delta $\delta^{\mu\nu}$ should always be of value $1$ if both indices are equal and $0$ if they are different. However it seems that the delta has components different from $1$ ...
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1 answer
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Covariant vs. contravariant definition of the Energy-Momentum tensor

I have a question regarding the definition of the energy-momentum tensor. I've seen it defined as a (2,0) tensor, so it has 2 upper indices $T^{ab}$, but many times it is written as a (0,2) tensor ...
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1 vote
1 answer
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Can the metric tensor be treated as a linear transformation?

In general relativity, the metric tensor $g$ is a covariant, second rank, symmetric tensor that can be written down as a 4x4 matrix. The metric tensor generalizes the notion of distance between points ...
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Proof of second Bianchi identity

I have a problem. I thought it was second Bianchi identity at first, but it's not. I have checked again and this formula is not wrong. How to prove it? $$R^n{}_{ikl;m}+R^n{}_{ikm;l}+R^n{}_{ilm;k}=0$$
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4 votes
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What is the physical motivation behind the mathematical definition of an inertial system?

In this German Classical Mechanics lecture by Frederic Schuller, it is given that a Newtonian spacetime with an absolute inertial frame is one in which $$ \nabla_{v} G=0$$ Where $\nabla_v$ is the ...
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4 votes
1 answer
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Argument of a scalar function to be invariant under Lorentz transformations

I'm trying to prove that a Lorentz scalar object $\rho(k)$ which is a function of a cuadri-vector $k^{\mu}$ can only have a $k^2$ dependency in the argument. I can imagine that this object has to ...
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1 vote
1 answer
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Preservation of symmetries of Tensors under lowering and raising indices

How do you go about showing that symmetry properties of tensors are preserved during lowering and raising indices in a metric space? I know how do do it for individual tensors with given symmetries ...
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0 answers
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Relativity Proof with Defining interval [duplicate]

I have started reading Landau & Lifshitz Vol. 2 (fields theory) and I've got confused about something I read. to prove the Lorentz transform, it defines interval: $$ds^{2} = c^{2} dt^{2} - dr^{2}\...
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5 votes
1 answer
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Why is a Lorentzian metric still Lorentzian after a general coordinate transformation?

In my GR course, we define a lorentzian metric $g_{\mu\nu}(x)$ as a symmetric $(0,2)$ tensor field having 3 positive and 1 negative eigenvalue. Now given a general coordinate transformation described ...
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Software recommendation for a tensor calculation

What is the best software/package to calculate $$2R_{\alpha\mu\beta\nu}R^{\mu\nu}-\nabla_\alpha\nabla_\beta R + \Box R_{\alpha\beta}-\frac12g_{\alpha\beta}\Big(R_{\mu\nu}R^{\mu\nu}-\Box R\Big)$$ for a ...
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Using embedding and identity or Minkowski metric instead of complicated metric [duplicate]

The Einstein's equation describes how the metric changes in presence of a mass. If we consider a spherically symmetric mass, the solution to it is given by the Schwarzschild metric. By complicated ...
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6 votes
3 answers
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What are the differential equations that model a self-propagating gravitational wave in space-time?

Light is a self-propagating wave, but it's very complicated. Imagine, if you will, a wave in space-time that by assumption was self-propagating like light, except that it was a gravitational wave. ...
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8 votes
1 answer
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How can an observer observe the metric of spacetime?

I don't mean how can we measure the metric in practice. I only mean in principle. Suppose you are an omnipresent being, no experimental limitations. What measurements do you need to measure the metric ...
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Equations of Motion and Minimization of Spacetime Interval

I'm trying to show that the extrema of a path in spacetime, as given by the spacetime interval (or length if just considering space) is the one that solves the equations of motion. Let a path be given ...
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3 votes
2 answers
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What's the physical content in the invariance of spacetime interval in GR?

Spacetime interval in one co-ordinate system is given by : $$g_{\mu \nu} dx^{\mu} dx^{\nu} \tag{1}$$ $dx$ is some infinitesimal displacement vector between two events. Spacetime interval after a ...
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