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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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How to determine if a tensor is covariant or contravariant?

In special relativity, the coordenates of a event are in general written using a 4-vector: $$x^{\mu} = \binom{ct}{\textbf{x}}$$ where $\textbf{x} = (x,y,z)$ are the spacial coordenates. This is a ...
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Difference between coordinate systame and frame of reference in relativity

I have taken a relativity course, and am not quite clear as to whether the notion of a frame of referene is independent of the coordinate system. At the moment, I think that any frame of reference (of ...
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How can time go in different directions in the Universe?

This is not a duplicate, I am not asking about any kind of time dilation caused by a BH. My question is about the direction of time (and if it is possible to have different directions for time, other ...
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1answer
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Sean Carroll GR - Ex.3.6 (b) & (c) [on hold]

I'm working in the newtonian limit of GR with the metric $$ ds^2 = -(1+2\Phi)dt^2 + (1-2\Phi)dr^2 +r^2d\theta^2+r^2sin^2\theta\;d\phi^2 $$ where $$\Phi = -\frac{GM}{r}.$$ We are first asked to ...
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Why is electric field $E^i = E_i$ instead of $E^i = - E_i$?

Let us consider the Minkowski spacetim. Generally, we know that when we lower or raise the index of the convariant or contravairant tensor, we need to use the metric $\eta^{\mu \nu}=\eta_{\mu \nu}=(+,...
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Tangent space of pseudo-Euclidean manifold

My textbook says (without any justification that I can see) that "since Minkowski spacetime is pseudo-Euclidean, the tangent space $T_P$ at any point $P$ coincides with the manifold itself". My ...
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52 views

Line Element Transformation

This is just something that I've made up to see if I understand the method. If I have the line element: $$ds^2 = dr^2 + r^2\,d\phi^2$$ and I want to carry out a transformation with $r = \dfrac{...
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Why is time 'negative' in a proper distance in Relativity? [duplicate]

so my question is as follows: the proper distance between two events $a$ and $b$, $S_{ab}$ is given by $$S_{ab}^2 = -[x_0(b)-x_0(a)]^2 +[x_1(b)-x_1(a)]^2 +[x_2(b)-x_2(a)]^2 +[x_3(b)-x_3(a)]^2.$$ ...
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Rotationally invariant metrics and conservation of angular momentum

This was prompted by an exam question, though the questions are more general: A 2D Riemannian space has the metric: $ds^2=dr^2 + \gamma^2 r^4 d\phi^2$ State what conserved quantity ...
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Momentum Operator on a Riemannian Manifold

Consider a non-linear sigma model on a Riemannian manifold with metric $g_{ij}$ with the action $$S= \frac{1}{2} \int dt g_{ij}(X) \frac{dX^i}{dt} \frac{dX^j}{dt}.$$ The momentum operator is $$P_i= \...
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Lorentz-transformation

I don't understand how to derive the matrix representing the Lorentz-transformation given two systems S and S': $$x' = \Lambda x$$ these transformations do not leave the differences $\Delta x^\mu$ ...
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Einstein Tensor doesn't vanish [on hold]

Hello my question is how is it possible that the Einstein tensor doesn't vanish in the Schwarzschild metric. For example $G_{11}= \frac{1}{2} g_{11} g^{22} g^{33} \partial_2 \partial_2g_{33}$ is one ...
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Rindler observers are at rest with respect to each other?

I'm studying a chapter about Rindler coordinates right now. In this they say that two Rindler observers at $x = 1/a_1$ and $x= 1/a_2$ will both have speed $v =0$ at $\tau = 0$ compared to the inertial ...
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2answers
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Metric tensor: Why relate it to Cartesian/Minkowski coordinates?

Why does the metric tensor always relate to cartesian coordinates? Let's take the simple case for the metric tensor in 3D-space without a time dimension, $g_{ij}= \begin{bmatrix} 1 & 0 &...
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Proving that test particles in GR, follow spacetime geodesics

My question is pretty much in the title. According to this paper, this is not exactly proven rigorously yet. What I dont understand is what exactly is not proven. If I'm not too wrong, a test particle ...
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Whats the quickest way to compute the Ricci tensor?

I have been going through exam papers and often they ask us to calculate ricci tensor components and affine connections from a given metric. They seem to take far too long for the time you are ...
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Radial infall in Schwarzschild

In Straumann's book on general relativity, one finds the following solution to the question of geodesic radial infall into the black hole: $$d\tau=(\frac{2m}{r}-\frac{2m}{R})^{-\frac{1}{2}}dr$$ To ...
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1answer
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Dual space and Metric tensor

So I know that the dual space is the set of all linear transformations that map a vector from a vector space to the field of the space itself (the real number line, complex, quaternions). From YouTube ...
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Proper time of an object crossing event horizon in Kruskal coordinates

So I am reading a paper on a certain black hole paradox. The specifics actually don't matter, but if you want context (p16): black hole thought experiment. An object falls into a black hole. The ...
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Imperfect or perfect fluid in Einstein Field Equation

I'm trying to solve the Einstein Field Equations in an unconventional way (at least not usual from what is done in most basic textbooks). So basically, I specified a metric tensor (specifying a ...
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1answer
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Is there a general relativistic mass in general relativity?

In general relativity the energy of a test-body moving in a spherically symmetric gravitational field can be written as: $$E=mc^2\left(\frac{\sqrt{1-\frac{2GM}{rc^2}}}{\sqrt{1-\frac{v^2}{c^2\left((1-\...
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Does the Einstein field equation uniquely determines the topology of spacetime? [duplicate]

I am trying to understand whether the Einstein field equation uniquely determines the topology of spacetime. As far as I know, given a metric, we can always find the induced topology. However, I was ...
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What is the formal difference between the light cone and a black hole?

A black hole can be loosely defined as a spatial closed surface from which nothing, not even light, can leave. The light cone of special relativity is in some sense similar to a black hole because by ...
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Penrose diagram and coordinate transformation

I am looking at the Minkowski line element in spherical coordinates $$ \mathrm{d}s^2 = - \mathrm{d}t^2 + \mathrm{d}r^2 + r^2 \mathrm{d}\theta^2 + r^2 \sin^2(\theta) \mathrm{d}\phi^2$$ and want to ...
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ADM formulation of GR derivative on the 3-metric

In the ADM formalism where the projector is given by ${P^\mu}_\alpha={\delta^\mu}_\alpha+n^\mu{n}_\alpha$ and $n^\alpha$ is a future pointing normal vector to the constant time hypersurface $\Sigma$. ...
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1answer
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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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1answer
265 views

Is a vector space automatically spacelike if it has a basis of spacelike vectors?

I am studying Kerr Spacetime and I am not sure about something used in a proof I am trying to understand. I am wondering, if you consider a 4-dimensional Lorentzian manifold $\mathcal{M}$ and $X_i \...
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2answers
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Derivation of the Electromagnetic Stress-Energy Tensor in Flat Space-time

I am working on deriving the electromagnetic stress energy tensor using the electromagnetic tensor in the $(-, +, +, +)$ sign convention. However, I have hit a snag and cannot figure out where I have ...
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1answer
66 views

Confusion Einstein notation polar coordinates

I'm having issues using Einstein notation in polar coordinates in flat space, I must be missing something basic. Consider the following example. Take the following metric on a 2+1 spacetime; $ds^2 = ...
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Curved space-time and metric tensor

I'm studying about curved spaces and I read that a manifold is flat if there a coordinate system such that the metric tensor is constant everywhere. Then I also read that when the space-time tensor ...
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How to show the metric is unchanged under an orthogonal transform?

Say we have the transform such that $x^i \rightarrow (x')^{i'}=M^{i'}_i(x)$ where $M$ is an orthogonal rotation matrix I've been asked to show that a general metric $g_{ij}(x)$ invariant under the ...
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1answer
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Properties of Dirac delta function in Integral

I was reading commutation relation of canonical momentum in KG Field from Lectures of Quantum Field Theory by Ashok Das. In page 179, He has used Integration to derive the result where he expressed ...
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1answer
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How to define time in a time-dependent solution?

If a spacetime has no timelike killing vector, how can we define "time" in such spacetime, in order to calculate the time evolution behaivor of some quantities in it?
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Basic Index Raising and Lowering Question [duplicate]

I am trying to understand the order of the indices when raising or lowering tensors. For example, the electromagnetic tensor: $$F^{\alpha \beta} = \begin{bmatrix} 0 & -\frac{E_{x}}{c} &...
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1answer
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Itzykson & Zuber: Conjugate momentum sign

I can't give myself peace on a confusion about the signs. I'm studying on Claude Itzykson & Jean-Bernard Zuber, Quantum Field Theory, Dover Publications. Metric convention $g_{\mu\nu}=diag(1,-1,-1,...
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2answers
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Electromagnetic Stress-Energy Tensor in curved space-time

I found on Wikipedia that the electromagnetic stress energy tensor in curved space-time with sign convention $(-, +, +, +)$ is $$T_{\mu\nu} = -\frac{1}{\mu_0} \left ( F_{\mu \alpha} g^{\alpha \beta} ...
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Isometry in two dimensions

Let A be an isometry, then $g^{ij}=A^i{}_{k}A^j{}_l\,g^{kl}$ (1) First, consider an infinitesimal transformation $A=\exp(\epsilon \lambda )$ Then express A by its power series to the first ...
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Proof that the scalar curvature of a two-dimensional space can be expressed by only one component of the Riemann tensor

So I'm working on a question that asks for a proof that in two-dimensional space, the scalar curvature is given by: $$R = \frac{2R^1{}_{212}}{{g}_{22}}$$ Now, I've been playing around with the ...
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What do orbits around a black hole look like doing the calculations in isotropic coordinates?

JPL and others calculating ephemerides in the solar system are using a method based on taking the Schwarzschild solution, not as expressed in the most common Schwarzschild coordinates but in the ...
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1answer
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Equality between derivatives of the metric

In one of my lecture, it is said: Let us use the freedom of the choice of parametrization to demand that the variation of $\lambda$ after a small displacement along the curve is proportional to the ...
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1answer
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General Relativity - (numerically) compute the metric from the stress-energy tensor?

I am new to GR and I am having trouble understanding how one goes back and forth between the metric $g_{\mu\nu}$ and the stress-energy tensor $T_{\mu\nu}$. First, have a look at the following post. ...
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Why isn't the scalar product of a covector and vector symmetric? [migrated]

In tensor math, how come the scalar product of a co-vector (co-variant vector) with contra-variant vector, as written between angle bracket separated by comma, $\langle x, a \rangle$, is not symmetric?...
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1answer
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Killing tensor and Conserved quantities

The definition of the killing tensor is written above, as taken from Wikipedia. My question here is two-fold: Can all Killing tensors be build from the Killing vectors of that spacetime? Do Killing ...
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1answer
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Questions about special relativity, index in the Lorentz matrix

I'm studying special relativity I have read this: We have $ x^u = (ct, x^1,x^2,x^3) $. If we apply Lorentz transformation we can write: $x'^u = \Lambda^{u}_{\hspace{0,2 cm}\nu} x^{\nu} $ $x'_u =...
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1answer
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Intuitive explanation for the Lorentz transformation for time

I've recently started learning SR, and while the Lorentz transformation for space is pretty obvious, just the Galilean transformation combined with space contraction, I can't figure out the ...
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2answers
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Is the determinant of metric tensor stationary wrt. proper time for a particle moving along its world line?

While writing the expression for stress energy tensor of a free massive particle moving along its world-line some authors take out of the integral sign, the $\sqrt{-g}$ where $g$ is the metric tensor ...
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2answers
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The Electromagetic Tensor and Minkowski Metric Sign Convention

I am trying to figure out how to switch between Minkowski metric tensor sign conventions of (+, -, -, -) to (-, +, +, +) for the electromagnetic tensor $F^{\alpha \beta}$. For the convention of (+, -,...
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2answers
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Is the concept “space” actually needed?

I started making my mind around space and time and recently came to a point where I wondered if the concept of "space" is actually needed to describe physical processes at all and not just some ...
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4answers
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Why does nature favour the Laplacian?

The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
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1answer
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Momentum of a moving object in FRW metric according to an observer comoving with cosmic expansion

I would like to show that in an FRW metric the momentum of a freely falling object decays as the inverse of the scale factor. I know there are many proofs and arguments for this but I am trying to get ...