New answers tagged general-relativity
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Is the equation $g_{\mu\nu} = const. + T_{\mu\nu}$ equivalent to Einstein's field equations?
Well, it simply fail to calculate correctly the gravitational field around a planet or star. The right side of the equation is zero in this case, but the space-time is not Minkowskian as your equation ...
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Is a Klein-bottle-like topology allowed for GR?
Observations suggest that our Universe is spatially homogeneous and isotropic.
This means that at a manifold level, the underlying spacetime should have 6 independent spatial Killing vector fields ($...
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Relationship between spacelike and timelike distances in General Relativity vs. Special Relativity
In Minkowski spacetime, the distance $d_S$ between two space-like separated events $x$ and $y$ [...]
"Distance" $d_S$ ??
Perhaps you mean the (spatial) separation between two (space-like) ...
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Is a Klein-bottle-like topology allowed for GR?
General relativity does not care about topology of the manifold. You could for example take a flat spacetime with cubical edges and identify the edges in such a way that you have a non-orientable ...
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Gravitational convergence of light
if space can change due to matter then close to a star light is different than that far from it?
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What would happen if the aether did exist and there was no such thing as relativity?
Interesting question. Brings to mind some thoughts.
After Maxwell published his theory of electromagnetism, many people thought that the EM waves had to be traveling through some medium, so they ...
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Relationship between spacelike and timelike distances in General Relativity vs. Special Relativity
The method to do this in special relativity is essentially the radar synchronization trick : the points $x$ and $y$ are being measured by some observer along the line $\overline{zw}$, such that it ...
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Can a body escape a black hole by being thrusted?
A less technical answer: a black hole is defined as an object that has a horizon from which nothing can escape from the inside to the outside. This also holds for light, which is the fastest anything ...
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Is the equation $g_{\mu\nu} = const. + T_{\mu\nu}$ equivalent to Einstein's field equations?
My guess is that the OP means $G_{\mu \nu}$ (the Einstein tensor) instead of $g_{\mu \nu}$ (which is the metric tensor). If that's the case then according to OP we should have
$$ G_{\mu \nu}=T_{\mu \...
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Test-particles motion in general space-time with *torsion* : geodesics or auto-parallel curves?
Without prejudicing the issue on either side, we can follow the Gandalf / Toucan Sam Strategy and follow our nose.
The matter can be addressed by applying the continuity equation to the stress tensor ...
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Questions about E. Minguzzi's article on Synchronization (arXiv:1009.3005)
Thank you for the interest in the paper. Let me mention that this work has not been published so far because soon after I posted it I worked on another version that expanded it while rearranging some ...
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Can a body escape a black hole by being thrusted?
The quick answer is no because time and spatial coordinates twist roles. As well as you can only move in one direction in time (forwards), inside a black hole you can't go backward in the radial ...
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The linear growth factor of perturbations today should be 1 but I am unable to derive it
That expression for the growth factor is not normalized to 1 today. Instead it is normalized such that
$$D(a)=a$$
during the matter epoch (i.e. before dark energy becomes important). This is another ...
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Is the "capacity to do work" of a body equivalent to the concept of "resistance to change in motion"?
The premise isn't quite correct; we could write $E=\sqrt{(mc^2)^2+(pc)^2}+V+E_0$, where $p$ is the momentum, $V$ is potential energy, and $E_0$ is a constant that sets the reference zero.
At small ...
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Modified Lie bracket
The reason to modify the lie bracket is the field dependence of $\zeta^\mu$.
For the full Einstein theory, the remaining parameters $\zeta^\mu$ persevering both gauge fixing and boundary conditions ...
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Wormhole Metrics and the Density of Negative Energy
Since I have worked with wormholes, modified gravity and all that (unfortunately), let me give you a somewhat mid-sized explanation of what these things are.
The point of energy conditions is to make ...
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Wormhole Metrics and the Density of Negative Energy
My first question to this site poses many sub-questions. However, I am now in a position to answer each of them to my satisfaction, and wish to share my results with the community. The generic answer ...
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Hypersurface orthogonal congruences
In Minkowski space, an exploding bag of clocks travelling radially at all speeds from the origin, determines a space-time hyperbolic, spatial spherical coordinate system in the future light cone by ...
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Twin paradox - how much energy does it take to travel to the future?
Energy is a frame-variant quantity, while the difference in the age between the two twins is an invariant quantity. An invariant quantity cannot be a function of only a frame-variant quantity, so ...
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Explanation about black holes
It's like running on a treadmill at maximum speed. It's such treadmill that it's wide, and it's curved longitudinally, so that if you slow down, you drift backwards, were your maximum speed is lower ...
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Is it possible to describe every possible spacetime in Cartesian coordinates?
I think you have a mixup between global and local properties here. Using differential geometry language, a space-time is a 4-dimensional manifold with a Lorentzian metric.
It is a theorem in ...
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Is it possible to describe every possible spacetime in Cartesian coordinates?
You can choose arbitary coordinates but GR tensor algebra is based on general covariance so that 4D curvature is preserved even if you change your frame of reference. See: https://en.wikipedia.org/...
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Why don't the Christoffel symbols transform as a tensor?
The clearest way to address the question is to treat the connection as a functional of vector fields $u = u^μ ∂_μ$ and $v = v^ν ∂_ν$ and a one-form field $α = α_ρ dx^ρ$, but to denote it as $Γ_{uv}^α$,...
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What vacuum should be defined for a observer in Kerr spacetime?
It depends on the physical situation, not on the observer. Let me give three examples. I will focus on the analogy with Schwarzschild spacetime, which has the Boulware and Unruh vacua, but the Hartle–...
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Do Cauchy horizons in AdS have a dual picture in the dual CFT?
This is interesting, see some general references below. To answer your question first:
Yes -- I am not aware of the exact dual picture, but as far as I understand one can use the holographic No ...
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Allowed Topologies for General Relativity
The theory by no means is only local.
The global degrees of freedom are determined by the curvature invariants integrated over all space. In other words if the local observables are determined ...
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Why can't the answers to equations be infinity?
No. Anyone saying that an "infinity is a way of telling you have made a mistake" is being too playful with words. Usually, infinities are a way of telling that you have done something ...
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When doing general relativity in practice, how do we choose the appropriate manifold describing the scenario?
As I understand your question, you are asking - "What local effects if any, are present in GR using which we can distinguish between manifolds (representing spacetime) that are same locally but ...
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Is it possible to describe every possible spacetime in Cartesian coordinates?
Since the question makes no reference to the number of dimensions, you could ask it just as well for a universe that is 2-dimensions of space and 1 of time. If you can't do it even there, then the ...
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Is it possible to describe every possible spacetime in Cartesian coordinates?
If OP by Cartesian coordinates means a local coordinate system $(x^0,x^1,x^2,x^3)$ [say, in some local open neighborhood $U\subseteq M$ of spacetime] such that the components $g_{\mu\nu}$ of the ...
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The limit of GR with infinite speed of light $c$
Well, you see there's a problem there. The actual kinematic symmetry group for non-relativistic physics is not the Galilei group, but the Bargmann group - its central extension. This is best seen by ...
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Is it possible to describe every possible spacetime in Cartesian coordinates?
Why would you want to write it in Cartesian coordinates? Putting aside everything, the statement "... using Cartesian Coordinates we could easier think about the structure of spacetime itself.&...
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Is it possible to describe every possible spacetime in Cartesian coordinates?
As the choice of coordinates is arbitrary, can't I just "postulate" to use cartesian coordinates to describe any possible spacetime?
If by cartesian coordinates you mean a set of four ...
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The limit of GR with infinite speed of light $c$
what would the universe be like if gravity was curvature but c was infinite?
The equivalence principle holds in Newtonian gravity. So you can geometrize standard Newtonian gravity.
That is called ...
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Equivalent condition for conjugate points (Wald's "General Relativity" Chapter 9.3)
Incidentally, I just saw this question. I would recommend Hawking and Ellis, since most of the concepts discussed are very mathematically clear and rigorous, albeit in a thick bold print. Not sure ...
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Hypersurface orthogonal congruences
From Hawking and Ellis, see the following discussion from proposition 4.5.1 and lemma 4.5.2. If $\mathcal{U}$ is a convex normal neighbourhood around some $p$, the points that can be reached from $p$ ...
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Does solving Einstein's field equation depend on Newtonian equations?
To fix the value of the coupling coefficient $κ$ that appears in Einstein's equation $G_{μν} = κ T_{μν}$ and connect it to the constant $G$ that appears in Newton's law of gravity, you need the ...
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Allowed Topologies for General Relativity
The theory only deals with the local curvatures, not the global topology. Hence any manifold with an allowed metric is allowed. These can be infinitely many, especially for negative curvature space-...
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Could black holes be a three dimensional object breaking through space time and falling 4th dimensionaly
Before you ask this kind of question it's a good idea to have some idea of the math behind Relativity.
The distance between two points on a plane is:
$$ds^2 = dx^2 + dy^2$$
Should be obvious - this is ...
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According the theory of general relativity, what is the role of causality in the changes of the curvature of spacetime?
Yes. This has to do with energy conditions and how the causal structure of the spacetime are linked. While the causal structure is usually somewhat axiomatic, in that the causal principle and ...
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How does the Penrose diagram for a spinning black hole differ in realistic scenarios (formed by stellar collapse)?
Since there are some good responses, let me put my two cents on a particular point you mentioned, with the inner Cauchy horizon $\mathcal{CH}_{I}$ and strong cosmic censorship.
In an "ideal" ...
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Inclination angle of a ray in a static spacetime
The inclination in a flat 3-space with metric
$$ d s^2=4 m^2\left[d \rho^2+\rho^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right)\right] $$
will be proportional to the coefficient of $d\rho$ divided ...
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How do changes in energy distribution update the curvature of spacetime?
You are thinking about gravity incorrectly. A planet does not send out gravitational signals. Nothing is emitted by the planet and the planet will never run out of energy by causing gravity. All ...
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No hair theorem and Klein-Gordon equation
What you are talking about seems to be a perturbation. You have a scalar field that exists in the Schwarzchild background.
No hair theorem is about back reaction.
You impose that the scalar field ...
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No hair theorem and Klein-Gordon equation
There is no singularity at the event horizon, its a perfectly smooth boundary, a light cone, that in its inner future and past has time terminating hyperboloids tangent to the cone in future.
In the ...
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Schwarzschild line element in Eddington-Finkelstein coordinates
What your are encountering here is a core feature of general relativity: time is inherently, and fundamentally a local quantity. While each observer will have an unambiguously defined local proper ...
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Why does this derivation of the Einstein Field Equations only work with the Trace-Reversed form?
You have the wrong dimensions for a few things, and these are arising from errors that have propagated widely even in the Physics literature. Among other things, it leads to the wrong value for $κ$. ...
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Christoffel symbol / scalar product of co- and contravarient basis vectors
I am going to summarize what @ghoster and @basics have said in the comments. As Ghoster points out, $$e^l\cdot \Gamma ^k_{nm}e_k = \Gamma ^k_{nm} (e^l \cdot e_k)$$ We know that, by construction, of ...
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Explanation about black holes
The idea of this has to do with what is referred to as the "expansion" of null (light) rays. When the expansion $\theta $ is zero, the two light rays will just be parallel to each other, ...
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Time Dilation And Comparing Inertial And Non Inertial Reference Frames
Total time dilation factor is the time dilation factor caused by the spin motion, measured in the frame where the spinning thing has no linear motion, multiplied by the time dilation factor caused by ...
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