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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76 views

Metric tensor times its inverse using Kronecker delta

From tensor calculus, we know that \begin{equation} g^{\mu\nu}=\delta_{\lambda}^{\mu}\delta_{\phi}^{\nu}g^{\lambda\phi}.\tag{1} \end{equation} Based on (1), is the following true? \begin{equation} g^{\...
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Are all maximally symmetric spacetimes conformally flat? What about the converse?

If I'm not mistaken, one of the properties of maximally symmetric spacetimes is that the Riemann tensor can be written as $R_{abcd} = \frac{R}{d(d-1)}(g_{ac}g_{bd} - g_{ad}g_{bc})$, which would imply ...
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What is the determinant of the induced metric $h_{ab}$ of the NG action?

In the introductory section of Polchinski String Theory: An introduction to the bosonic string we are given the induced metric which reads $$ h_{ab} = \partial_a X^{\mu}\partial_b X_{\mu}\tag{1.2.8} $$...
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Is the space and time dimensions in Schwarzschild metric orthogonal?

Schwarzschild metric does not contain any cross-terms such as $dtd\phi$, does that mean that space and time coordinates are orthogonal to each other? Should the dot product of any time vector with ...
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Are isometries really global symmetries?

On one hand, a spacetime $(M,g)$ with the Killing vector $\xi^\mu$ and $x^\mu(\tau)$ a geodesic, we can construct the quantity $$Q = \xi_\mu \frac{dx^\mu}{d\tau}\tag{4.32}$$ that is constant along the ...
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2D-metric to diagonal form with determinant 1 [closed]

I wonder whether it is always possible to bring a 2D Riemannian metric to a diagonal form with determinant one by changing the coordinates, i.e. for the line element $$ ds^2 = A(x,y)\, dx^2 + B(x,...
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Present value of the cosmological scale factor, important or arbitrary?

I'm now confused about a subject that I thought was very clear to me, until recently. So I need to sort this clearly once and for all. Consider the standard FLRW cosmology in classical general ...
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1answer
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Geodesic incompleteness of static spherically symmetric solution

Static spherically symmetric solution of Einstein equations is given by the metric $$ ds^2=f(r)dt^2-\frac{dr^2}{f(r)}-r^2d\Omega^2, $$ where $f(r)=1-(kr)^2$, $d\Omega^2$ is the metric of unit sphere. ...
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What does the $t$ coordinate represent in a general metric?

I am learning general relativity, I understand that the metric tensor has a coordinate $t$ corresponding to time. But I know also that time depends on gravity and so the time can change from point to ...
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Question on the first fundamental form

The first fundamental form is related to the metric tensor of the manifold as follows: $$h_{ab} = g_{ab} - \sigma \text{ } n_a n_b$$ Where $\sigma$ is +1 or -1 depending on normalization of the normal ...
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Help understanding Einstein notation

This is basically the same question as this one. I have the same problem with the sign. In the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$, the term $i\gamma^{\mu}\partial_{\mu}$ is: $$i\...
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Event horizon from a metric in Cartesian coordinates

How can we derive the event horizon from a metric dependant on $(t,x,y,z)$? I've seen the Schwarzschild solution and the Kerr solution, but both of these are given in $(t,r,θ,φ)$. I tried converting ...
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How are these two different definitions of covariant vector related?

Definition 1 If under a coordinate transformation $x^i\to \bar{x}^i(x^i)$ certain objects $A^i$ transform as $$A^i\to \bar{A}^{i}=\sum_{j}\frac{\partial \bar{x}^{i}}{\partial x^j}A^j,$$ those objects ...
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Interpretation of four-vectors on a manifold

I have a couple of silly questions related to the concept of manifold. In Euclidean or flat Minkowski space I can easily specify an origin $O$, and then describe the position of a particle at a point $...
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Generalization of conformal mapping of a metric along the line of yellow CFT book

In the book CFT by Francesco chapter$-5$ while deriving the conformal maps in $2$ dimension i.e. $5.2,5.3$ or simply $z \rightarrow w(z)$. The author's assumed the initial metric to be $$g_{\mu\nu}=\...
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Where do we get our sense of perpendicularity?

We all have an innate sense of perpendicularity. I know, for example, that if I make a T-pose then the direction of my arms and the direction of where my face is pointing towards will be perpendicular....
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How can we physically determine the metric in SR? How can we physically determine a coordinate system where $g$ is diagonalized?

Here I'll use theoretical units with $c=1$. Suppose we are given two coordinate systems $(t, x)$ and $(\tilde{t}, \tilde{x})$, and suppose they are related by $t = \tilde{t} - \tilde{x}/2$ and $x = \...
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Computing curvature singularities from a metric

Suppose I have the metric $$ds^2 = f(r)(dt^2-dr^2-dz^2) - \frac{1}{f(r)} d\phi^2. $$ How would you calculate the curvature singularities of this metric if we assume that $f(r)$ takes value $0$ for $...
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How are units handled general relativity? How are units handled in curvilinear coordinate systems?

I'm having trouble understanding how the units of distance are handled when we think of (non-Cartesian) coordinates on a spacetime manifold. All sources I know of seem to brush this question under the ...
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Is the Palatini formulation of a gravity theory a metric one?

As far as i know, the Einstein equivalence principle (EEP) is the starting point to explain gravity as a geometric phenomenom. It allows you to link gravity with two geometrical objects: a metric, ...
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Inferring features about the spacetime from a line element

Suppose $g_{ij}$ is a symmetric tensor describing the geometry of a 4d spacetime. Then I know for example, that if the metric contains "mixed terms" (like $dtdx$) then this spacetime is not ...
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Showing that null geodesics are incomplete

Given a metric: $$ds^2 = -dt^2 + t dx^2$$ for a manifold $M = \mathbb{R}^+ \times \mathbb{R}$. The geodesic equation for null geodesics is $$x = x_0 \pm \log t$$ for some constant $x_0$. Now I want to ...
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Manipulation of Einstein's equation

In my lecture notes, there is the following consequence of Einstein's equations, (it follows on to have multiple consequences, so I don't think there's a mistake in the notes): $$R_{\mu\nu} - \frac{1}{...
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Is the line element a distance vector or displacement vector?

In my Electrodynamics and Electromagnetism course, the professor is deriving Maxwell equations and the electromagnetic field tensor from differential geometry and wants to show how special relativity ...
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What would happen if a black hole disappeared? [closed]

Imagine if a black hole disappeared. Would spacetime act like a rubber band and propel objects that used to be caught in its gravitational field outwards - i.e. some kind of space time explosion? How ...
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Is a static spacetime always spherically symmetric?

I'm a bit confused. In this question it is suggested that a static spacetime can be spherically asymmetric. A static spacetime is one for which the metric doesn't change in time. It's irrotational too....
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Carroll's GR: The Schwarzschild Metric

On page 195 of Sean M Carroll's An Introduction to Special Relativity: Spacetime and Geometry, he calculates the Christoffels and Riemann tensors for a generic metric that is static and has spherical ...
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Derivation of Bondi metric

I was trying to understand the BMS formalism from the beginning. The whole formalism depends on the expression of the Bondi metric. I don't understand how this expression of the metric was derived. ...
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Why constant metric tensor allows finite displacement

Consider as space coordinates $q_1,q_2,q_3$ why if the metric tensor is not constant in space $q_1, q_2, q_3$ can't be considered a displacement vector? And why vice versa, a constant metric tensor in ...
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1answer
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Normalization of 4-velocity using an arbitrary parameter (not proper time)

The normalization of the 4-velocity vector is \begin{equation} g_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=-1 \end{equation} I understand that if we parametrize the curve with some arbitrary ...
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Proof that 4-velocity is normalized in curved spacetime

Whenever I try to find an explanation for the normalization of the four-velocity \begin{equation} g_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=-1 \end{equation} I'm always shown a proof in ...
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Is there a contradiction in the description of the 4-vector french page of wikipedia?

In french wikipedia of 4-vector : https://fr.wikipedia.org/wiki/Quadrivecteur#De_la_base_covariante_aux_quadrivecteurs_covariants The description starts with an explanation, in context of special ...
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Energy-momentum tensor of a fluid for scalar fields

I know that the energy-momentum tensor for a perfect fluid in General Relativity is given by $$T_{\alpha \beta} = (\rho +p)u_{\alpha}u_{\beta}-p\,g_{\alpha \beta}, $$ where $\rho$ is the density, $p$ ...
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Proper distance and proper time in general relativity

I consider a worldline $x^{\mu}(\lambda)$, where the parameter $\lambda$ parameterizes the world line. Consider now the distance between the two points $x^{\mu}(\lambda+d\lambda)$ and $x^{\mu}(\lambda)...
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What can be derived from the metric tensor? [closed]

I am working on a computational project about General Relativity. In this process, I want to code 'the stuff' that can be derivable from the metric tensor. So far, I have coded Riemann Tensor, Weyl ...
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1answer
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Hamilton-Jacobi-Einstein equation

I have been looking at the Hamiltonian formalism of GR for some time and recently stumbled across the Hamilton-Jacobi-Einstein equation: $$\frac{1}{\sqrt{g}} (\frac{1}{2}g_{pq}g_{rs} - g_{pr}g_{qs}) \...
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Locally flat coordinate on Poincaré half plane

I have a question from " Einstein gravity in a nutshell by A. Zee", chapter I.6 question 4: " Find the locally flat coordinate on the Poincaré half plane." Also, there is a hint ...
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Is it possible for coordinate distance to disagree with proper distance in flat spacetime?

The title pretty much sums it up. I'm learning about general relativity and I've ran across descriptions of coordinate distance not matching with a moving objects proper distance. We know this is ...
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Why is the Schwarzschild metric equal to 0 for light at the event horizon?

Carroll & Ostlie's Introduction to Modern Astronomy and Astrophysics provides a derivation on why light and information is frozen at the event horizon: However I noticed that they equated $ds=0$ ...
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Which is the conformal factor of $S^2$?

I'm confused about Conformal factor. All 2D riemannian manifolds are conformally flat. That means, if $g$ is the metric, exists a scalar function $\Omega$ that $g_{a,b}=\Omega^2 \delta_{a,b}$ In the ...
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1answer
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Trying to solve Einstein's Field equation , i.e Finding each components of the equation

Does knowing the $FRW$ metric equation is required first so as to solve Einstein's field equation ? $\Rightarrow$ First FRW, then the christoffel symbols, then the riemann tensor, then the ricci ...
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Null geodesic correct Lagrangian

I am aware that the Lagrangian for a relativistic massless particle is different than that for a massive particle, as the usual action (dots denote derivative w.r.t. $\lambda$) $$S =m\int d\lambda \...
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Determine tetrad of local inertial frame numerically

Let $g_{\mu\nu}$ be a metric of a curved spacetime. However, we do not know its analytical form. We only know $g_{\mu\nu}$ and its derivatives from a former numerical simulation and the underlying ...
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Dirac equation more natural in $O(1,3)$ than $O(3,1)$?

In physics, the Lorentz group $O(1,3)$ is of central importance, being the setting for electromagnetism and special relativity. According to Wikipedia: Some texts use $O(3,1)$ for the Lorentz group; ...
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Can we modify classical total energy equation to get desired results (Schwarzschild metric results) for planetary orbits in a Lagrangian analysis?

(A) In a three-dimensional flat space, total energy $H = T + U$ in a conservative field is constant. The three-dimensional Lagrangian ($L = T - U$) gives force on an object (see image- eq. (1)). $T$ ...
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In the context of condensed matter physics, what does it mean for time to have two dimensions?

In an online article that describes condensed matter physics for laypersons, the author describes various so-called "designer materials" that have exotic properties, including one in which ...
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Nonlinear extension of Lorentz Group [duplicate]

The Lorentz group is defined to be the set of linear transformations that leave $ds^2 = -dt^2 + |d\vec{x}|^2$ invariant. The Poincaré group contains the Lorentz group, but now we allow transformations ...
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The metric exterior of a massive object

The only condition apart from perfect spherical symmetry that is required for the retrieval of the Schwarzschild-metric $g_{ik}$ is actually ($R_{ik}$ being the Ricci tensor, the contraction of the ...
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1answer
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Different signatures of the metric in Einstein field equations

Throughout the GR lectures, we have always used (- , + , + , +) signature for the metric tensor but in some chapters it was switched to (+ , - , - , -) and immediately after that Einstein field ...
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Tensor contraction and covariant derivative

I have naive question about GR and Covariant derivative. You know \begin{align} \nabla_{\gamma} g_{\alpha \beta}=\nabla_{\gamma} g^{\alpha \beta}=0 \end{align} And I would like to compute covariant ...

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