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Where is the Lorentz signature enforced in general relativity?

4-dimensional Lorenzian manifold was the most natural way to express the invariance of light velocity (Maxwell's wave equation), in terms of metric $$ds^2=c^2 dt^2-dx^2-dy^2-dz^2$$ For Herman ...
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-1 votes

Curvature of Hilbert space

Hilbert space is infinite dimensional space, by default it is continuous, has no curvature, and extends indefinitely in all directions. It also lacks any edges where the space ends, or wraps around on ...
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Where is the Lorentz signature enforced in general relativity?

The Lorentzian signature of the metric is "baked into" the local causal structure (the set of "light cones", one at each event) of spacetime, which plays a role in the initial ...
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Where is the Lorentz signature enforced in general relativity?

The Lorentz signature is just part of the theory; for example in a weak-field limit, we should reduce to special relativity, which is described using Lorentz signature (in order to talk about light, ...
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4 votes

Where is the Lorentz signature enforced in general relativity?

Quoted from Pseudo-Riemannian manifold - Lorentzian manifold: A principal premise of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold of signature $(-,+,+,+)$...
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2 votes

Are there types of spacetime that have no symmetries?

Let me rephrase your question. You are asking if there exists a Riemannian (or a Lorentzian) manifold $(M,g)$, such that any smooth vector field $X$ on $M$ satisfying the Lie derivative equation $$\...
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4 votes

Are there types of spacetime that have no symmetries?

I assume you're actually asking about isometries of the metric $\phi^* g = g$ in the context of GR (physical spacetime symmetries). The 'gauge symmetry' of GR, diffeomorphism invariance, is an ...
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Hyperbolic geometry in SR

I'd just like to justify John Rennie's claim that the hyperbola is the set of points where the point can boosted to: In an $x-y$ plane, a Lorentz boost is defined as: $$ x' = \gamma(x-vy)$$ $$ y' = \...
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1 vote

Could time be a secondary effect due to curvature of space?

Are there cases where time is curved independent from space? To my knowledge not. Einstein field equations (EFE) are just about this dependency. In case of Schwarzschild exterior solution the metric ...
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3 votes

Could time be a secondary effect due to curvature of space?

You can't have curvature in 1 dimension: it is a line and always locally flat, with $R^a{}_{bcd} = R^0{}_{000} =0$. So it doesn't make sense to ask about 'time being curved'. If you want to consider ...
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Could 2D spacetime be seen as an embedded manifold?

If you mean "embedded in $\mathbb{R}^3$ with a Euclidean metric", then the answer is no. Suppose that such an embedding exists. Consider the neighborhood of a point $P$ on the submanifold. ...
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Linear Momentum in General Relativity

Is there a metric that describes this case? Yes. It is the Schwarzschild metric (valid outside of the gravitating body if we are talking about something like a star). When written in the form where ...
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Linear Momentum in General Relativity

Already answered on physics.stackexchange.com here. Likewise mentioned on Wikipedia Looks like the answer is "yes".
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How to simplify the process of calculating spacetime geodesics?

I guess you may be making a reasonable point here, when speaking about this particular metric. Assume the space-time is split into $(t , \, x) \, \in \, \mathbb{R} \times M_3$, where $M_3$ is one of ...
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How do we figure out what is the right geometry of space?

We are used to call Newton's law of gravity to the equation:$$\mathbf a = -\frac{GM\mathbf r}{r^3}$$ for a point in the vacuum at a distance $r$ from the center of a spherical body. Taking $\mathbf a =...
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4 votes

How do we figure out what is the right geometry of space?

Don't we we already have a geometry of space time as soon as we write down the Minkowski metric? Yes, but that's putting the cart before the horse. Einstein's equations are a system of differential ...
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How to calculate Proper Distance as an arc length in Schwarzschild metric?

For a fixed $r$ and $t$, $$ds^2 = r^2(d\theta^2 +\sin^2\theta\ d\phi^2)\ ,$$ which is the same arc length as in Euclidean space. Then if $\phi = f(\theta)$ then $$s =r \int \left(1 + \sin^2\theta\ \...
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Does the stress energy tensor scale with the metric tensor?

To supplement Eletie's answer, note that under the transformation $g\mapsto \lambda g$ with $\lambda$ an $\mathbb R$-valued constant, the Christoffel symbols remain unchanged because $\Gamma \sim g^{-...
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2 votes

Does the stress energy tensor scale with the metric tensor?

What you're asking about is a specific case of Weyl transformations of the form $$ g_{ab} \rightarrow e^{-2\phi(x)}g_{ab} $$ where $\phi$ is a constant. Under these, the Ricci scalar is not invariant (...
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Change of Metric Under Coordinate Transformation

I think that you are trying to compute the Lie derivative of the metric. If so, there should be no factor of 1/2. Under an infinitesimal shift $x^\mu\to x^\mu +\eta^\mu$ we have $g \to g+ {\mathcal ...
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How can I show that the inverse of the induced metric $h_{\alpha \beta}$ is $h^{\alpha \beta}$?

In differential geometry, $h$ is the pullback of the spacetime metric $g$ along the immersion of the worldsheet. Thus it is automatically a metric and writing this in local coordinates will give the ...
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-1 votes

How can time go in different directions in the Universe?

If time represents all change in the current state of the universe, then any change in the relationship of variables in spacetime could be interpreted as a change in the direction of time. Example - ...
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4 votes
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Riemann curvature tensor in an inertial frame

The fact that a function's first derivative vanishes at a point does not mean that its second derivative vanishes at that point. Note that for $f(x)=x^2$, $f'(0)=0$ but $f''(0)=2$.
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3 votes

Invariance of spacetime interval by Schutz

I would suggest you read Landau & Lifshitz argument for invariance of $ds^2$, particularly the Wikipedia link because it states clearly the theorem which is being proved, and it fills in a few ...
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3 votes
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Invariance of spacetime interval by Schutz

The claim is that if the relationship between coordinates in the two inertial frames is linear and if $\Delta S^2 = 0$ necessarily implies that $\Delta S'^2 = 0$, then in general we must have that $\...
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2 votes

Understanding EFE: RHS linear, LHS not?

This is just a small comment to buttress Andrew's answer (since it was a long string of comments on his answer!). Let us review the definitions of basic geometric gadgets used in Einstein's field ...
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11 votes

What is the evidence that gravitational fields don't sum up as a superposition?

Black hole solutions would not exist in a linear theory of gravity. This is because black holes are vacuum solutions, not sourced by any matter, and there are no static vacuum solutions that die off ...
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What is the evidence that gravitational fields don't sum up as a superposition?

Gravitational wave (GW) observations of binary black holes (BH) may provide experimental tests of superposition of spacetimes, as defined by you. Each BH is described by a spacetime metric, but the ...
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6 votes

What is the evidence that gravitational fields don't sum up as a superposition?

Einstein's equations are $$ G_{\mu\nu}[g] = R_{\mu\nu}[g] - \frac{1}{2} g_{\mu\nu}R[g] = 8\pi G_N T_{\mu\nu} \tag{1}. $$ where $g_{\mu\nu}$ is the metric of the spacetime. The Ricci scalar is given by ...
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Trying to derive newtonian potential for Schwarzschild interior metric

To get the equation (10.54) you should integrate from $\Phi(r)$ to $\Phi(R)$ and not from $\Phi(0)$ to $\Phi(R)$. The same relates to the integral on the right side of your equation which has to be ...
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Why is it necessary that different observers agree on the value of the spacetime interval $ds^2$?

space intervals in Newtonian mechanics In Newtonian mechanics different observers can disagree on the position of events. As an example, let's say I am $100$ m to the left of you. An event, $A$, ...
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2 votes

Physical significance of metric compatibility

Metric compatibility follows from the Einstein equivalence principle, which asserts that the laws of special relativity hold for local non-gravitational phenomena in a freely falling laboratory. Here, ...
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Why is it necessary that different observers agree on the value of the spacetime interval $ds^2$?

FIRST POSTULATE OF SPECIAL RELATIVITY The laws of physics are the same and can be stated in their simplest form in all inertial frames of reference. SECOND POSTULATE OF SPECIAL RELATIVITY The speed of ...
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Physical significance of metric compatibility

There is a formulation of GR where metric compatibility follows from the equations of motion. If you think of the Einstein-Hilbert action as being a functional of both the connection and the metric (...
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3 votes

Understanding EFE: RHS linear, LHS not?

Consider the equation \begin{equation} x^2 + 1 = x \end{equation} The left hand side is nonlinear in $x$, while the right hand side is linear, but this is still a valid equation. Actually the ...
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2 votes
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Is this version of Einstein field equations linear?

Einstein's equations are not linear in the metric tensor $g_{\mu\nu}$, which is the field we want to solve for$^\dagger$. This means that if $g^{(1)}_{\mu\nu}$ and $g^{(2)}_{\mu\nu}$ are two solutions ...
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I'm confused about the number of Killing vectors in Schwarzschild metric

A brute force (and ugly) derivation of the Killing fields of Schwarzschild metric The Schwarzschild metric is \begin{equation} ds^2 = -\left(1-\frac{R_{\text{S}}}{r}\right) \text{d} t^2 + \left(1-\...
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1 vote

Finding the proportionality constant in $\varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho}\propto \varepsilon^{\lambda\rho}$

The properties of the epsilon tensor (as exploited by Cosmas Zachos) give a very elegant answer here. But, if you didn't know about those identities, you can still get the answer in a straightforward ...
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2 votes
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Finding the proportionality constant in $\varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho}\propto \varepsilon^{\lambda\rho}$

Just use the standard properties of the tensor in solving for C, $$ \varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho} = C \varepsilon^{\lambda\rho} ~~~~~\leadsto \\ \epsilon _{\lambda \rho} \...
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Change of coordinates from an arbitrary frame to a locally inertial frame in General Relativity

I have looked at the three-dimensional analogue of the problem Michael treats. Their are 27 equations involved, of the form \begin{equation*} g_{ \rho\sigma,\lambda }=a~(b_{\sigma\lambda\rho}+b_{\rho\...
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Metric tensor relations

The first definition on the first line is identical to the definition on the second line, $Q_{\alpha} = g^{\mu \nu} Q_{\alpha \mu \nu}$. (I'm not sure what the $ْ\ $ is doing there, perhaps this is ...
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2 votes
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Components of the fully contravariant Kronecker Delta in Schwarzschild Metric

The index structure of the Kronecker delta is $\delta^\mu_\nu$, not $\delta_{\mu\nu}$ OR $\delta^{\mu\nu}$. In ANY coordinate system, $\delta^\mu_\nu$ has value 1 if both indices are equal and 0 if ...
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Covariant vs. contravariant definition of the Energy-Momentum tensor

The metric $g_{\mu\nu}$ can be used to raise and lower indices. In this case you are lowering the indices on the SEM tensor, that is converting it from contravariant to covariant. So the answer is ...
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Can the metric tensor be treated as a linear transformation?

The metric can be thought of as a linear map between two vector spaces, but they're not the same vector space. Rather, we can think of the metric as a map from $V \to V^*$, the dual space of the ...
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1 vote

What is meant when it is said that the universe is homogeneous and isotropic?

Most of modern cosmology is based on the Cosmological Principle, which states that the spatial distribution of matter in the Universe is homogeneous and isotropic when viewed at a sufficiently large ...
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