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 much Gravity is required to stop time?

Clocks free of gravitational influence run faster than those experiencing gravity. Is it possible for gravitational influence to bring time to a stop? Additionally can acceleration affect clocks in ...
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865 views

Orbits around the Photon sphere of a black hole (Schwarzschild coordinates)

This is a follow-up question to the answer given at What is the exact gravitational force between two masses including relativistic effects?. Unfortunately the author hasn't been online for a few ...
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1answer
58 views

Spacetime background of Quantum mechanics

Why is it said that the Schrodinger equation suggests a fixed, non-dynamical background spacetime, with time as an external parameter? How does this interpretation come about from the Schrodinger ...
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1answer
39 views

Why are dimensions regarded as square/perpendicular?

Starting from the second dimension, the dimensions are basically represented by a square, cube, tesseract, and so on. I don't know if this is a stupid question or not, but is there an obvious or ...
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2answers
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Do any two points in Minkowski spacetime determine a unique line?

Any two points in a Euclidean space determine a unique line, but I wasn't sure if this result generalized to Minkowski spacetime given that the latter is not a Euclidean 4-space, but is, instead, a ...
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0answers
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Scalar Curvature of a Conformally Flat Metric

Suppose that you have a metric $g_{\mu\nu}=\phi^2\eta_{\mu\nu}$ for some function $\phi$. There is a standard formula for what the scalar curvature $R$ looks like in terms of $\phi$, which is given by ...
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4answers
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Space-like and time-like: where do the names come from?

Space-like separated events are events that, in a well-chosen reference frame, can take place at the same time but never happen at the same location. On the other hand for time-like events, one can ...
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2answers
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What is the difference between time and space in general relativity?

I know that similar questions have been asked before, I will try to be specific. In special relativity time is the coordinate with minus sign in metric tensor. In general relativity the components of ...
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0answers
26 views

Coordinate time difference between emiting and detecting a photon in bent spacetime

Consider an arbitrary non-trivial metric $g_{ij}$ - like the Schwarzschild metric. Now, consider two observers $A$ and $B$, staying at fixed radii $R_A$ and $R_B$, respectively, with $R_A > R_B$. ...
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1answer
228 views

How can I mathematically describe the parallel transport in the Roman soldiers example?

I've been trying to understand parallel transport. Many of the descriptions present a mathematical version: $\nabla_V X = 0$. And/or they present an example involving soldiers (usually Roman) ...
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1answer
62 views

Are the Schwarzschild metric and the Geodesic Equation relevant in the context of the Earth? [closed]

The geodesic equation used in general relativity is the following: $$ {\mathrm d^2 x^\mu \over \mathrm ds^2} =- \Gamma^\mu {}_{\alpha \beta}{\mathrm d x^\alpha \over\mathrm ds}{\mathrm d x^\beta ...
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1answer
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Signature of $f: \Lambda^2(\mathbb{R}^4) \times \Lambda^2(\mathbb{R}^4) \to \mathbb{R}$, $f(\omega, \omega') = \omega \wedge \omega'$ [closed]

Define$$f: \Lambda^2(\mathbb{R}^4) \times \Lambda^2(\mathbb{R}^4) \to \Lambda^4(\mathbb{R}^4) \cong \mathbb{R}, \quad f(\omega, \omega') = \omega \wedge \omega'.$$ What is the signature of $f$? ...
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What spacelike, timelike and lightlike really mean?

Suppose we have two events $(x_1,y_1,z_1,t_1)$ and $(x_2,y_2,z_2,t_2)$, then we can define $$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$ which is called the spacetime ...
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1answer
257 views

Are we traveling through time at the speed of light? [duplicate]

In the image below the y axis represents time and x represent velocity. Point D represents velocity c, point E represents 1 second per stationary observers second. What this chart is showing is as you ...
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1answer
95 views

Calculating speed in four dimensions [closed]

If you are moving at $c$ in 3D space and $c$ in time axis too, What would be your total speed? Edit: Since question has been voted to be closed, I shall make an Edit. In 4D world all objects move ...
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Induced metric is a scalar for transformation from $x\to x'$? (Poisson E.A p.62)

I have a (simple) question about the induced metric $h_{ab}$. In Poisson E.A. (a relativist toolkit) it says in p. 62 that the induced metric $$h_{ab}=g_{{\alpha}{\beta}} \frac{\partial ...
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1answer
64 views

How do you actually use the geodesic equation?

The geodesic equation used in general relativity is the following: $$ {d^2 x^\mu \over ds^2} =- \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}. $$ It states that the ...
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1answer
31 views

Lie Derivative of Kahler 2-form

Suppose there is a Killing vector $k$ on a Kahler manifold $M$. By definition, $k$ generates isometries of the metric. That is, $L_kg=0$, where $L$ is the Lie derivative. At the same time, there is a ...
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1answer
42 views

Physical reasons for metric definition in special relativity [duplicate]

I am working through "General Relativity" by Wald, and am currently going through the brief section on Special Relativity. The spacetime metric is defined as $\eta_{ab} = \sum\limits_{\mu, \nu=0}^3 ...
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2answers
125 views

(Hyper)Surface of Simultaneity

How can I determine the surfaces of simultaneity if I know the metric? In particular, what are the surfaces of simultaneity for rotating disk with Langevin metric: $$ ...
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1answer
250 views

Null geodesics in uniform gravitational field metric

I'm trying to understand the null geodesics in the metric: $$\mathrm{d}s^2 = -(1+gz)^2 \mathrm{d}t^2 + \mathrm{d}z^2 + \mathrm{d}x^2$$ In particular I'm wondering if the following intuition is ...
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2answers
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How does one measure space-like geodesics? Or: What is the physical interpretation of space-like geodesics?

In general relativity, time-like geodesics are the trajectories of free-falling test particles, parametrized by proper time. Thus, they are easy to interpret in physical terms and are easy to measure ...
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2answers
68 views

Trouble understanding spacetime and invariant interval

First, how is the invariant interval useful? How can it help us understand things around us in the universe? Second, I know that they changed time into space or better say SPACETIME in order to ...
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How can we recover the Newtonian gravitational potential from the metric of general relativity?

The Newtonian description of gravity can be formulated in terms of a potential function $\phi$ whose partial derivatives give the acceleration: ...
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1answer
75 views

Metric that is Minkowski plus sum of null vectors

In GR exercises I've often seen metrics of the form $g_{ab} = \eta_{ab} + k_ak_b$ where $k_a$ is null with respect to $g$ (or equivalently $\eta$). I'm happy doing calculations with such metrics, but ...
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1answer
95 views

Trying to understand Newtonian limit of GR

First ever post - please be kind. I'm trying to understand how General Relativity becomes equivalent to Newton's laws of motion, plus Newton's law of gravitational attraction in the limiting case of ...
4
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1answer
51 views

Schwarzschild metric, acceleration of ball before it's dropped [duplicate]

The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - 2M/r)^{-1}\,dr^2 + r^2(d\theta^2 + ...
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2answers
210 views

Performing Wick Rotation to get Euclidean action of scalar field

I'm working with the signature $(+,-,-,-)$ and with a Minkowski space-stime Lagrangian $$ \mathcal{L}_M = \Psi^\dagger\left(i\partial_0 + \frac{\nabla^2}{2m}\right)\Psi $$ The Minkowski action is $$ ...
4
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2answers
108 views

For a giving metric in GR, how do we learn which observer the metric refer to?

For example, I have been told the Schwarzschild observer is far away from blackhole and events,(namely, I think, the observer is static at infinity of the coordinate.) And the second example,the ...
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4answers
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The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
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1answer
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How can you tell if spherical-like coordinates are locally flat across the origin?

In general relativity, with spherical-like coordinates in a radial gauge, I have a metric that looks like: $$-g_{tt}\mathrm{d}t^2 + g_{rr}\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\ ...
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2answers
72 views

Deriving $A^{\mu}_{;\nu}$ from $A_{\mu ; \nu}$

We have a covariant derivative of a covariant tensor: $$ A_{\mu ; \nu} = A_{\mu , \nu} - \Gamma^{\alpha}_{\mu \nu} A_{\alpha} $$ The covariant derivative of a contravariant tensor is: $$ ...
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1answer
45 views

Norm of the momentum 4-vector

The norm of the momentum 4-vector is $\mathbf{P}.\mathbf{P}$ $= (\gamma mc, \gamma mv).(\gamma mc, \gamma mv) = \gamma mc^2 - \gamma mv^2$ But why is $\gamma mc^2 - \gamma mv^2 = mc^2$?
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1answer
117 views

What is the metric of a constant electromagnetic (pure electric or pure magnetic) field?

For example, imagine a magnetic field $B_x$ directing in $\hat{x}$ direction filling all the space. What is its associated metric field? I can construct the electromagnetic stress-energy tensor for ...
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2answers
75 views

Straight line null geodesics in Minkowski, De Sitter and Schwarzschild

I'm trying to understand which part of the following metric determines whether photons travel on a "straight" line (thinking of $(t,r,\theta,\phi)$ as a flat background), the metric I'm considering ...
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0answers
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Volume form of the AdS_{4} Space

Regarding the unit radius $AdS_{4}$ space, the metric in global coordinates, is given by: $$ds^{2}_{AdS_{4}}=\frac{1}{\cos^{2}{\rho}}[dt^{2}-d\rho^{2}-\sin^{2}\rho d\Omega_{2}^{2}]$$ where ...
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2answers
228 views

Coordinate Singularity in Metric

Suppose I have some metric $$ds^2=g(t)dt^2+\frac{1}{r}dr^2$$ which has a singularity at $r=0$. However, if I make the coordinate transformation $u=\frac{1}{r}$, then I get: $$ds^2=g(t)dt^2+r^3 ...
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30 views

AdS boundary global vs Poincare'

Is the global boundary of AdS the same of the boundary written in Poincare' coordinates?
79
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1answer
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Is there such thing as imaginary time dilation?

When I was doing research on General Relativity, I found Einstein's equation for Gravitational Time Dilation. I discovered that when you plugged in a large enough value for $M$ (around $10^{19}$ ...
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Gravitational lensing and cosmic strings

Say we have a straight cosmic string lying along the $z$-axis, with energy-momentum tensor $$T_{\mu\nu}=\mu\delta(x)\delta(y)\operatorname{diag}(1,0,0,-1)\tag{1}\label{1}$$ for some small positive ...
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4answers
168 views

How do we know the Schwarzschild solution contains an object of mass $M$?

The Schwarzschild metric is $$ds^2 = - \left( 1 - \frac{2GM}{r} \right) dt^2 + \left(1-\frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2.$$ In Carroll's GR book, it is claimed that $M$ is the mass of the ...
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6answers
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Proving that interval preserving transformations are linear

In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity: Prove ...
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Motivation for usage of 4-vectors in special relativity

I understand that if one considers a 4-dimensional space-time from the outset then 4-vectors are the natural quantities to consider (as opposed to 3-vectors as in Newtonian mechanics), since the ...
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2answers
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Definition of the Lorentz transformations [closed]

Until very recently I believed that the Lorentz transformations were defined as "the transformations that carry one inertial reference frame into another". In Wikipedia's page we find something along ...
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1answer
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Homogeneity and isotropy and derivation of the Lorentz transformations

In deriving the Lorentz transformations I have found (from reading a few different sets lecture notes) that it is argued that they must be linear and thus there general form must be $$x'=Ax+Bt,\quad ...
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Inertial coordinate systems [duplicate]

In Newtonian mechanics, by the following two assumptions: (i) The time is absolute. (ii) The length is absolute. it is easy find the relations betweem two coordinate systems with uniform motion ...
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2answers
189 views

What manifold is spacetime?

In General Relativity, spacetime is a $4$-dimensional manifold with one Lorentzian metric tensor defined on it. In the Special Relativity case what manifold is spacetime is quite clear: it is ...
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1answer
126 views

Diffeomorphism invariance and geodesic action

I'm trying to understand the role of diffeomorphism and isometry invariance in the geodesic action in GR: $$ S = \int_{\tau_1}^{\tau_2} \! d\tau~ g_{ab}(x(\tau)) \frac{dx^a}{d\tau} \frac{dx^a}{d\tau} ...
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1answer
56 views

Questions about null geodesic [closed]

Show for the null geodesic in 3D flat spacetime using polar coordinates so the line element is $ds^2=-dt^2+dr^2+r^2d\phi^2$. Do light rays move on straight lines? My question is that I only learned ...
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Interpreting meaning of coordinates given a metric

I was working problem 3.6 in Carroll's GR textbook and was given the following metric, which is a good approximation to the metric outside the surface of the Earth. $ds^2=-(1+2 \Phi(r))dt^2 + ...