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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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deSitter spacetime metric and curvature

I have to compute the metric of an hyperboloid given by $-(X^0)^2+(X^1)^2+(X^2)^2+(X^3)^2=H^{-2}$ in 5D Minkowski spacetime using the following coordinates: $$X^0=H^{-1}\sinh(Ht)\sqrt{1-H^2r^2}$$ $$X^...
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1answer
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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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Proving two space-time intervals are equivalent with matrix algebra

η=$ \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $ v=$ \begin{bmatrix} ct\\ x\\ y\\ z \end{...
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2answers
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Spacelike and timelike intervals confusion

I'm confused about this, specifically the spacetime interval. A timelike interval is one in which 2 events can be related to each other in a given reference frame within its light cone, that is, it ...
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1answer
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How is four-velocity automatically normalized?

This is a page from Sean Carroll's Spacetime and Geometry.There is a line in this page which says that the four velocity is automatically normalized.This absolute normalization is a reflection of the ...
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1answer
64 views

Refraction of matter waves

My question is a bit messy, so here is the background: Normally the trajectory of a massive particle in the presence of a gravitational field is described in the context of general relativity. ...
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How to find Ricci tensor?

I'm trying to find the Ricci tensor in question 3. Here $u=r/R .$ http://imgur.com/gallery/qSAknvz I found the Christoffel symbols but I can't find the Ricci tensors. On the link, there is also my ...
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Are all 4D Ricci flat manifolds locally Euclidean? [closed]

If a 4D manifold with metric signature (++++) is Ricci flat. Does this mean that locally the space is Euclidean? Does this mean that the only difference between these manifolds is there global ...
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2answers
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Paradox are all 4D distances zero?

The 4 dimension distance from the origin of a point is $\sqrt{x^2+y^2+z^2-t^2}$. Which means the 4 dimensional distance on the light-cone is zero. Take a point A and a point B in the future at ...
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11answers
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Is spacetime wholly a mathematical construct and not a real thing? [closed]

Speaking of what I understood, spacetime is three dimensions of space and one of time. Now, if we look at general relativity, spacetime is generally reckoned as a 'fabric'. So my question is, whether ...
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Varying a scalar field Lagrangian density

I was varying a scalar field density and I look at this term $${\cal L}~=~-\frac{1}{2}\partial _\mu\phi\partial^\mu\phi.$$ The result that I need to come is $$-\frac{1}{2}\delta(\partial _\mu\phi\...
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Why Empty universe have to obey the Negative Curvature? [duplicate]

For empty universe it seems to me that we can have two solutions. $$H^2=\frac {8\pi G\epsilon} {3c^2}-\frac {\kappa c^2} {R^2a^2(t)}$$ For an empty universe when we set $\epsilon=0$ we get $$H^2=\...
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1answer
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Will I be destroying spherical symmetry if I write the mass of the body as a function of time?

Will I be destroying spherical symmetry if I write the mass of the body as a function of time? If yes, then how can I write a metric for a body with mass as a function of time?
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Non-static spherical symmetry spacetime

The Schwarzschild solution is a static spherically symmetric metric. But I wanted to know that how would the space-time interval look in a Non-Static case. I tried to work it out and got $$ds²= Bdt² -...
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1answer
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$\partial^{\nu} \partial_{\nu}$ vs. $\partial_{\nu} \partial^{\nu}$

I was doing a problem regarding field theory. I am given the following lagrangian density: $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi_i\partial^\mu\phi_i-\frac{m^2}{2}\phi_i\phi_i$$ for three scalar ...
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1answer
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Line element 1-form

It was pointed out that dual vectors of a manifold, and hence differential 1-forms, are not dependent on the metric (Intuition behind dual vectors ('Bongs of a bell' does not help)). But doesn'...
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Visualization of $ dtdx$ and $dxdy$ term in metric tensor

For the sake of simplicity, lets take a 2+1 dimensional spacetime. Lets take the metric $$ds^2 = g_{tt}dt^2 + g_{xx}dx^2 + g_{yy}dy^2 + g_{tx}dtdx + g_{xy}dxdy$$ What is the visualization or ...
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3answers
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Intuition behind dual vectors ('Bongs of a bell' does not help)

Similar to the post here (How to visualize the gradient as a one-form?), I'm wondering about an intuition behind dual vectors and differential forms (and the link in that answer to Thorne's notes is ...
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1answer
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Direction of gravity

General Relativity explains the path a falling body makes (ex. An apple falling toward the center of the Earth) as a geodesic in curved spacetime. What explains the direction the apple falls? In other ...
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Why does a sign difference between space and time lead to time that only flows forward?

Ever since special relativity we've had this equation that puts time and space on an equal footing: $ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$ But they're obviously not equivalent, because there's a sign ...
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Robertson-Walker metric and cosmic homogeneity

The Robertson-Walker metric is of the form $$\tag{1} ds^2 = dt^2 - a(t)^2 \Big(\frac{dr^2}{1 - kr^2} + r^2 d\theta^2 + r^2 \sin^2\theta \, d\phi^2 \Big).$$ My question is related to the $a^2(t)$ ...
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General Relativity: Exchanging a field with its infinitesimal components on metric tensor

On a youtube video about Einstein's field equations, the author writes the following equation (https://youtu.be/foRPKAKZWx8?t=1078): $$d\phi=\sum_{n} \frac{\partial \phi}{\partial x^n} dx^n\tag{1}$$ ...
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2answers
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I almost got the Schwarzschild solution [closed]

I was trying to derive the Schwarzschild metric, but i got the following solution: $$ds^2=(C_1-\frac{1}{r})dt^2-\frac{1}{(C_1-\frac{1}{r})}dr^2-r^2d\Omega^2.$$ The correct place for the $C_1$ should ...
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2answers
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The time component of the geodesic equation for Newtonian gravity

I am working on a simple and popular GR textbook exercise. In Dodelson's Modern Cosmology (p. 54), it is stated thus: The metric for a particle traveling in the presence of a gravitational field ...
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Proper distance in cosmology

If considering two galaxies $\mathcal{A}$ and $\mathcal{B}$ being separated by $d$ ($d$ being the comoving distance) and light is emmited from $\mathcal{A}$ at time $t_1$ and that light is received to ...
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0answers
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Fiber manifold Ricci flat, physical meaning

In a warped-product spacetime, what a physical meaning we have for Ricci-flat Fiber? I'll explain.. it is well known that a Ricci-flat spacetime means that the cosmological constant need not vanish, ...
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Geodesic curve definition [duplicate]

Do we have a choice in defining the covariant derivative by the use of a set of coefficient functions(Christoffel gammas)? If so, could we then say that these coefficient functions need not to ...
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1answer
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Sign mistake in the energy momentum tensor of the Klein-Gordon Equation

Recently I understood that the energy momentum tensor can be calculated by: \begin{equation} T_{\mu \nu}=\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{\mu \nu}}.\tag{1} \end{equation} So consider ...
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1answer
64 views

Calculus of Variations help [closed]

I've been studying Chapter 6 in Taylor's Mechanics book. And am working through the odd-numbered problems. I am struggling with 6.13, which reads: In relativity theory, velocities can be ...
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2answers
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Is it spacetime or space that is expanding?

I have read answers to several similar questions but I still don't get it. Earlier explanations seem to say that laws of quantum physics and general relative are different. Let me get this straight. ...
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How can one obtain the metric tensor numerically?

I am self-studying General Relativity. Is there a method for obtaining the metric tensor exterior to a specified mass distribution numerically? In the simplest case of a spherical mass this should ...
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1answer
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Radius of Star, The Schwarzschild metric and Black Holes

From Section 9.1, in General Relativity by Woodhouse: For a normal star, the Schwartzchild radius is well inside the star itself. As it is not in the vacuum region of space-time, the Ricci tensor ...
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1answer
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When can we raise lower indices on “nontensors” as described in Dirac's book *General Theory of Relativity*?

This is a follow on to my previous question: Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero? I should not have made that ...
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3answers
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I have a question regarding the Painlevé-Gullstrand (PG) metric with factor 2

I have a question regarding the Painlevé-Gullstrand (PG) metric. If we have the line element in a radial fall we get: $$d\theta = d\phi = 0$$ $$ds^2 = -dT^2 + \left(dr+\sqrt{\frac{r_s}{r}}dT\right)^...
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4answers
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Invariance of the relativistic interval

Imagine we have two events, $E_1, E_2$ in the coordinate systems $K, K'$ (with coordinates $(x,y,z,t),\ (x',y',z',t')$), whilst $K'$ ist moving with the speed $\vec v$ in regard to $K$. From the ...
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1answer
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Deriving the equation relating the metric and the coordinates to the proper time in general relativity

If $x^a = x^a(\tau)$ is the worldline of a particle in general motion, then $$V^a = \frac{dx^a}{d\tau}$$ is a four-vector field along the worldline. If $$g_{ab}V^aV^b = g_{ab}\dot{x}^a\dot{x}^b = ...
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Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero?

See the bold text for my question. This question regards Dirac's General Theory of Relativity page 13. In his demonstration that the length of a vector is unchanged by parallel displacement Dirac ...
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1answer
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Write down the components of metric tensor correctly [closed]

this is a FLRW metric and I want to write down the metric tensor from this FLRW metric accurately. Can anyone please help me to do this? Thanks in advance. \begin{equation}\tag{1} ds^2 = a^2 ( \tau) [...
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4answers
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Why was pseudo-Euclidean geometry not enough for general relativity?

How would you explain to someone the change that Einstein needed in geometry for his new ideas about gravity and spacetime, what did he seek but could not be described by pseudo-Euclidean geometry? ...
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PG-Metric dr/dt

at the moment, I investigate some time in learning General Relativity and there I saw the Painlevé-Gullstrand Metric which is given as $ds^2 = -dT^2 + \left(dr+\sqrt{\frac{r_s}{r}} dT\right)^2 + r^2 ...
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4answers
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Understanding the difference between timelike and spacelike separations

From Woodhouse's General Relativity: If $A$ is the origin and $B$ is a nearby event with coordinates $dt, dx, dy, dz$, then, $$ds^2 = dt^2 - dx^2 - dy^2 - dz^2$$ is the same in all local inertial ...
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Closed form expression for position as function of time of object falling directly into black hole from infinity

Given a Schwarzschild radius $r_s=2 G M/c^2$, the escape velocity (equal to speed if falling from infinity) will be $\sqrt{2 G M/r}=\sqrt{r_s c^2/r}$ where the radial distance "r" is the point at ...
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2answers
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Can we compose tetrads together to represent metrics with many sources?

I've taken to viewing tetrads as a linear transformations of the Minkowskian metric to some curved space. Really though I'm just using this as a device to picture their action (as it could be viewed ...
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Tidal acceleration of a body at rest in a Schwarzchild geometry using geodesic equations

I'm self-studying the properties of a Schwarzchild geometry, with line element $$ds^2 = \left(1-\frac{2m}{r}\right)dt^2-\left(1-\frac{2m}{r}\right)^{-1}dr^2-r^2\left(d\theta^2 + \sin^2\theta d\phi^2\...
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3answers
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Does time speed up or slow down near a black hole?

The Schwarzchild geometry is defined as $$ds^2=-\left(1-\frac{2GM}{r} \right)dt^2+\left(1-\frac{2GM}{r} \right)^{-1}dr^2+r^2(d\theta^2+\sin^2(\theta) d\phi^2)$$ Lets examine what happens close to ...
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1answer
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Can we always choose a gauge in GR in which time is constant?

In General relativity the metric describes the curvature of 4D space-time. But due to diffeomorphism invariance, many metrics describe the same physics. Can we always choose a metric such that we can ...
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1answer
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States of classical general relativity

In Classical Mechanics a state of a system is either a pair $(q,p)$ or $(q,\dot{q})$ depending if we formulate the theory on the tangent or cotangent bundle of the configuration space. The evolution ...
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1answer
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(3+1)D solution to (2+1)D einstein equations?

Imagine a grid in 3D made of pipes smoothed so that it forms one continuous infinite surface. The surface is 2D but it fills 3D space. Like this (at one instant): Could any surface like this be a ...
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Finding the Ricci tensor components for the Schwarzschild metric

I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the ...
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2answers
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Four-velocity and the metric tensor

The metric tensor $g_{{\mu}{\nu}}$ has this property $$g_{{\mu}{\nu}}g^{{\mu}{\nu}}=4$$ and the four-velocity, $U^{\mu}=\frac{dx^{\mu}}{d\tau}$ which has this property $$U_{\mu}U^{\mu}=c^2.$$ So ...