5 votes
Accepted

References on Newton-Cartan Gravity

First, the MTW book has a chapter 12 on Newton-Cartan (NC) theory, but it is rather outdated in the sense that some important results were obtained after this book was published. A book by D. ...
4 votes

When does Newtonian physics fail?

"Fail" is a relative term. For some applications a 1% error is enough to say the theory is failing, for others getting roughly the right order of magnitude is a success. Special relativity ...
Anders Sandberg's user avatar
2 votes

How does the conformal scaling behavior of the stress-energy tensor depend on the spacetime dimension?

This is just dimensional analysis. The Hamiltonian $$ H=\int\mathrm d^D\boldsymbol x \, T $$ has dimensions of energy, so $T$ has engineering dimension $\Delta=D+1$. So for example, in $d=1+1$, $\...
AccidentalFourierTransform's user avatar
2 votes

References on Newton-Cartan Gravity

I don’t have very many in-depth references, but, here’s a brief list: Elie Cartan - On Manifolds with Affine Connections and the Theory of General Relativity (original French in 1923, translated to ...
2 votes

The order of the time-space components of the metric tensor in post-Newtonian expansion

In the weak-field (Newtonian) limit, if you were to calculate the Einstein tensor and the Einstein equation, you would get: \begin{equation} \Delta ^{(2)}g_{00} = \Delta \phi = 4\pi G\rho \end{...
tchonky fritz's user avatar
2 votes

When does Newtonian physics fail?

Regarding kinematics, and kinetic energy ($T$): $$ T = E-mc^2 = \sqrt{((mv)c)^2 + (mc^2)^2} - mc^2 $$ So we with $\beta=v/c$ and doing the Taylor expansion around $\beta=0$: $$T =\Big( \frac 1 2 \beta^...
JEB's user avatar
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2 votes

When does Newtonian physics fail?

We have two completely different models of physics: Newton's, which is a very good approximation for everyday purposes, and Einstein's, which is an even better approximation and which is known to ...
niels nielsen's user avatar
2 votes

Geodesic in flat space in spherical coordinates

You have made some major mathematical mistakes. There is actually no contradiction here at all. You have only defined $\mathbf{u}$ to be the tangent vector to a curve in the manifold $M$, which means ...
Vincent Thacker's user avatar
1 vote

How does the conformal scaling behavior of the stress-energy tensor depend on the spacetime dimension?

Under Weyl transformations, the partition function of a CFT transforms as $$ Z[\Omega^2g] = e^{-S_{an}[\Omega,g]} Z[g] $$ where $S_{an}[\Omega,g]$ is the anomaly action. This is non-vanishing only in ...
Prahar's user avatar
  • 26k
1 vote
Accepted

Preferred state of motion from topology of certain spacetimes

Topology alone does not lead to existence of such “preferred state of motion”. For example, two dimensional de Sitter space has the same topology $E^1\times S^1$ yet it is a maximally symmetric ...
A.V.S.'s user avatar
  • 15.7k
1 vote

Obtaining TOV equations

You need to use Einstein's equations. They'll give relations between your metric functions ($e^{2\Phi}$, $e^{2\Lambda}$) and the energy momentum components that you'll be able to solve for $\Lambda$ ...
Yudi's user avatar
  • 31
1 vote

What would be an experimental test of Sciama’s theory and why it has not been pursued yet?

There is a discussion of Sciama's paper in Williams and Inan's paper Maxwellian mirages in general relativity, New J. Phys. 23 (2021) 053019: https://iopscience.iop.org/article/10.1088/1367-2630/...
alanf's user avatar
  • 7,482
1 vote

How to motivate that in presence of gravity the spacetime metric must be modified to $ds^2=g_{ab}(x)dx^adx^b$?

It's a consequence of the equivalence principle, for which a gravitational field is locally indiscernible from an acceleration effect, for any moving small body. And the corollary: a free fall in a ...
Cham's user avatar
  • 7,421
1 vote

Angular velocity of stationary observer

ZAMO stands for Zero Angular Momentum Observer. The angular momentum of a worldline in Boyer-Lindquist coordinates is exactly $u_\phi$. So for a ZAMO $u_\phi=0$ per definition. Since $$ u^\phi = u_t g^...
TimRias's user avatar
  • 11.6k
1 vote
Accepted

What made Einstein to think time also dilates, along with space, with increasing gravity?

This is a direct consequence of the Einstein equivalence principle, which states that being stationary in a uniform gravitational field $g$ is equivalent to being in a reference frame with proper ...
Vincent Thacker's user avatar
1 vote

What made Einstein to think time also dilates, along with space, with increasing gravity?

Gravitational time dilation comes out of the solutions to the Einstein field equations, ultimately. But a heuristic reason to expect it is that light rising in a gravitational field must lose energy, ...
Eric Smith's user avatar
  • 9,129
1 vote
Accepted

Is there a location in the universe with the minimum rate of time dilation?

The universe is, to the best of our current knowledge, both isotropic and homogenous at large scales. So there will be no one point that is singled out at cosmological scales. All that would matter is ...
Dale's user avatar
  • 100k
1 vote

Geodesic in flat space in spherical coordinates

The real question is, "what exactly is a divergence?" The physical meaning of the divergence of a point particle's geodesic doesn't make sense. A divergence is meaningful for a vector ...
Paul T.'s user avatar
  • 7,095

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