9
votes
Accepted
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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9
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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 ...
- 91.9k
7
votes
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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7
votes
Accepted
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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6
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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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4
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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.&...
- 355
3
votes
Accepted
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 ...
- 355
2
votes
The Weird Interpretation for Contravariant and Covariant Vector
You're almost there, you just have the reasoning reversed. Suppose that $x'^\alpha = 2 x^\alpha$. Then all of the contravariant components double, just as you said. But a vector is left invariant ...
- 99.9k
2
votes
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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2
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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 ...
- 276
2
votes
Accepted
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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2
votes
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 ...
- 91.9k
2
votes
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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1
vote
Accepted
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 ...
- 15.7k
1
vote
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 ...
- 358
1
vote
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 ...
- 330
1
vote
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 ...
- 355
1
vote
Accepted
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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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/...
1
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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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