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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Question on Painlevé-Gullstrand coordinates

A known set of coordinates used for the Schwarzschild metric is the Painlevé-Gullstrand coordinates. They consist in performing a change from coordinate time $t$ to the proper time $T$ of radially ...
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Spacetime diagram for static weak field metric

Can anyone draw/provide links to the spacetime diagram for the static weak field metric (where Newton's law applies)?
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Deriving Einstein-Rosen Bridge

I know that an einstein rosen bridge is derived by a coordinate transformation on the schwarzschild metric, but I can't find much on it online, could someone please show how to change the metric into ...
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Question on the Einstein-Hilbert action

Does it make sense to write that the Einstein-Hilbert action as \begin{equation} S=\int\mathcal{L}\left(g^{\mu\nu},\partial^{\lambda}g^{\mu\nu}\right)\sqrt{-g}\,\mathrm{d}^4x=\frac{1}{2\kappa}\int R\,\...
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Components of a vector after translation

If you have a vector expressed in curvilinear coordinates and you translate it to a different point in space, the components of the vector will change to ensure that it maintains its geometric ...
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How does the metric change under scale transformations?

I was reading Zee's book in Group Theory, chapter VIII.2 (The Conformal Algebra) and I'm stuck in something probably easy: In p.516, he states that under a scale transformation, $x'^{\mu} = \lambda x^...
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Objective direction of time in general relativity

In general relativity, the coordinates $x^\mu$ we put on the manifold are arbitrary and need not have any physical interpretation. For this reason, it is said there is no objective notion of time in ...
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Does Pythagoras theorem hold for length contraction according to Special relativity?

Here is a thought experiment. Consider a frame S in which an observer is at rest and she looks ahead to find a right angle triangle of sides 3 (x-axis),4(y-axis) and hypotenuse 5. Now consider another ...
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Weak field metric derivation from the equivalence principle

Can weak field metric be derived just from the equivalence principle? I mean both the spatial and the temporal parts. Please provide a derivation if possible.
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$\displaystyle{\not}{a}\displaystyle{\not}{a} = a^2$ or $-a^2$ in Srednicki

I'm confused: In Srednickis Book (Equation 37.26), he has: $$\displaystyle{\not}{a}\displaystyle{\not}{a} = -a^2$$ However, every other source I found (for example this SE question says that it's: $$\...
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Extrinsic curvature calculation

In this paper section 3.2 the author uses the second junction condition to derive equation 3.8 $$ K_{1ab} = \frac{1}{L_1}\tanh\left(\frac{\rho_1^*}{L_1}\right)h_{ab}\;\;\;\;\;\;\;\;\;\;\;K_{2ab} = -\...
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Problem in Tensor calculation

I'm a theoretical physics M.SC. student, I have some problems in extending a tensor. Can anyone open up this tensor? it will be a great help for me. And also I want to know what is the difference ...
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Confusion regarding divergencies of the Kretchmann scalar for the time dependent Schwarzschild metric

First of all, I know the Vaidya metric exists and I know the properties of the Ricci scalar/Ricci tensor for this metric. But The metric I use is the following: \begin{equation} ds^2=\left( 1-\frac{...
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Landau Classical Fields theory argument for invariance of $ds^2$

In Landau's classical field theory, chapter 1, I'm getting mad with an argument used to derive the $ds^2$invariance from Einstein's postulate of the invariance of the speed of light. Landau says that ...
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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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On the definition of comoving coordinates

In chapter 8 Carroll's Introduction to General Relativity: Spacetime and Geometry, he defines a set of coordinates to be 'comoving' if the metric is free of cross terms $dt\,du^i$ and the coefficient ...
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Why in the curved space-time, the double derivatives of the position vector is symmetric but any other vector is not symmetric?

The double derivatives of the position vector (see image eq. (1)), connecting the two points in a curved space-time defined by the Schwarzschild metric, are symmetric under no torsion condition. This ...
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Metric coefficients in rotating coordinates

Let $(t,x,y,z)$ be the standard coordinates on $\mathbb{R}^4$ and consider the Minkowski metric $$ds^2 = -dt^2+dx^2+dy^2+dz^2.$$ I am trying to compute the metric coefficients under the change of ...
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Is the spatial metric that you get by writing the Minkowski Metric in a rotating coordinate system curved or not?

Suppose I am watching a rotating ring. My question is will its radius and circumference get contracted due to length contraction. My logic to solve the problem: Each small differential length of ...
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Weyl transformation of the metric

In this reference https://arxiv.org/abs/1809.09975 Ketov, et al. “Extending Starobinsky Inflationary Model in Gravity and Supergravity.” [1809.09975] Extending Starobinsky Inflationary Model in ...
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Metric independent gauge-fixing conditions for gauge theories

For instance, the Lorenz gauge conditions $$\partial^{\mu}A^a_{\mu} = 0$$ is implicitly dependent on the metric because of the raised index of the partial derivative. So, is there a metric independent ...
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Is it always possible to decompose the metric tensor of a spacetime using an arbitrary well-defined vector field ( or rather flow)?

In his book, A relativist's toolkit, in the section where Poisson is talking about the expansion and sheer vectors, he uses deviation vectors to decompose the metric into two parts. But to me, it ...
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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}{...
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Understanding the Plane Symmetric Metric

I don't understand as to what is the point of having a plane symmetric universe / metric at all? I mean shouldn't any physically sensical cosmological model (e.g. FLRW Model) entail a spherically ...
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Kaluza-Klein mechanism reparameterization $A'_\mu=A_\mu-\partial_\mu\lambda$

Polchinski String Theory volume 1 chapter 8 the parameterize of the metric in Kaluza-Klein mechanism was given by $$ds^2 =G_{\mu\nu} dx^\mu dx^\nu + G_{dd}(dx^d +A_\mu dx^\mu)^2$$ where $\mu,\nu\in [...
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Asymptotic expansion of curvature tensors in asymptotically flat spacetimes

Let us consider one asymptotically flat spacetime and consider a neighborhood of ${\cal I}^+$ with Bondi coordinates $(u,r,x^A)$ in which the metric takes the form $$ds^2=-du^2-2dudr+r^2\gamma_{AB}dx^...
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Tensor index in special relativity?

I'm studying special relativity and I have some difficulties with tensor index. Take for example the Lorentz matrix, whose elements are written as $\Lambda^\mu{}_\nu$. $\Lambda^\mu{}_\nu(v) = \...
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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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Is Birkhoff's Theorem Valid in Higher Dimensions?

I have seen a question here on PSE asking the same thing, but it has no answers. To reformulate the question slightly differently, suppose you have a general spherically symmetric metric in $(1+D)$ ...
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Conformal transformation vs diffeomorphisms

I am reading Di Francesco's "Conformal Field Theory" and in page 95 he defines a conformal transformation as a mapping $x \mapsto x'$ such that the metric is invariant up to scale: $$g'_{\mu \nu}(x'...
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Derivation of $zz$-component of Einsteins Equations in AdS

I am trying to understand how we get the Einsteins equations in here section 4.1 equation 4.2 where we use the metric $$ ds^2 = a^2(z)(dz^2+dx^\mu dx_\mu) $$ to derive the $zz$-component of Einstein's ...
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Riemannian metric induced by $n$ point sets in Hilbert spaces?

Excuse my naive question: In quantum mechanics the mathematical description is through Hilbert spaces. Suppose we have $n$ points $x_1,\cdots,x_n$ in this Hilbert-Space. Then we get the Gramian matrix:...
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How to get the four-velocity components from a given metric tensor?

I’m a little bit confused about how to get the four-velocity components from a given metric tensor (or line element). For instance, which are the components of the four-velocity in the Schwarzschild ...
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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 ...
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Shift Vector in Warp Equation

In the Alcubierre metric, why is there a beta with subscript multiplied by a beta with superscript? I know beta with subscript is the shift vector, but what is the difference between the two? $$\text ...
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Zamolodchikov metric and coupling-dependent field rescalings

this is my first post/question, so I apologize in advance for any mistakes in phrasing or format or anything else. Consider a conformal field theory (CFT) with exactly marginal deformations described ...
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D'Alembertian for a scalar field

I have read that the D'Alembertian for a scalar field is $$ \Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu). $$ Exactly when is this correct? Only for $...
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What is the physical meaning of the Eddington-Finkelstein coordinates?

What is the physical meaning of the Eddington-Finkelstein coordinates? I want to see a some physical process (experimental) that could explain the many transformations of coordinates into this ...
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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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Newtonian Limit of Schwarzschild metric

The Schwarzschild metric describes the gravity of a spherically symmetric mass $M$ in spherical coordinates: $$ds^2 =-\left(1-\frac{2GM}{c^2r}\right)c^2 \, dt^2+\left(1-\frac{2GM}{c^2r}\right)^{-1}dr^...
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Gaining intuition over Kruskal Extension of Schwarschild Metric

The Schwarzschild metric is relatively easy to visualize even though the metric is singular at $r=2M$ and $r=0$. Once we make the Kruskal extension, we find whole new regions in the manifold including ...
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Proper time in General Relativity and change of coordinates

Let $M$ be the spacetime manifold and let us consider a local coordinate system \begin{align} \varphi_i:\,U_i&\subset M\to \varphi_i(U_i)\subset \mathbb R^n, \end{align} which associates $p\in ...
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Perturbation of the energy-momentum tensor: mistake in my computations or in the book?

In the book Cosmology - S. Weinberg eqauation $(5.1.28)$ is $$ \delta T^{\mu}_{\;\;\nu} = \bar{g}^{\mu\lambda} [\delta T_{\lambda\nu} - h_{\lambda\kappa} \bar{T}^{\kappa}_{\;\;\nu} ] \tag{5.1.28} $$ ...
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Confusion with certain aspects of the Equivalence Principle

I took a course on GR and I am revising it after a while. I am heavily confused about the equivalence principle. Consider the following two statements: On a Riemannian manifold, we can always choose ...
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Tidal Love numbers and the Schwarzschild metric

I'm puzzled by the statement below: Consider the Einstein equation expanded to the linear order around the Schwarzschild background. This describes perturbation of black hole solution $$g_{\mu\nu}=g_{\...
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Geometrical representation of Contravariant and covariant vectors

After cruising through a lot of material online, and answers over here, my understanding of contravariant and covariant vectors are, in a finite-dimensional vector space, suppose we have a vector, ...
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Fixing non-calculus proof that Lorentz transformations are linear

Define the Lorentz group to be $$O(1,3)=\{\Lambda:\mathbb{R}^4\rightarrow\mathbb{R}^4|\eta(\Lambda u,\Lambda v)=\eta(u,v)\},$$ where $\eta$ is the Minkowski inner product. One could try to mimic the ...
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Computing the metric tensor from its Killing vectors?

On page. 139 of Carroll's GR book, during the discussion of Killing vectors, he quotes an explicit coordinate basis representation for the Killing vectors on $S^2$: \begin{array}{l} R=\partial_{\phi} \...
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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 ...
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Non-diagonal elements of the Schwarzchild metric

The Schwarzchild metric is the most general spherically symmetric, vacuum solution of the Einstein field equations. I was wondering if there was a simple argument to explain why the Schwarzchild ...

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