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Can solutions of GR have non-zero genus?
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Imagine there is "cavity" in one's locally Lorentzian manifold (the manifold has non-zero genus). Have these kind of solutions in general relativity been considered? If so, where can I read more about this? Is there any experiment one can do to prove we live in a non-zero genus world?
general-relativity differential-geometry topology specific-reference
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up vote
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Imagine there is "cavity" in one's locally Lorentzian manifold (the manifold has non-zero genus). Have these kind of solutions in general relativity been considered? If so, where can I read more about this? Is there any experiment one can do to prove we live in a non-zero genus world?
general-relativity differential-geometry topology specific-reference
asked 3 hours ago
More Anonymous
967
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Imagine there is "cavity" in one's locally Lorentzian manifold (the manifold has non-zero genus). Have these kind of solutions in general relativity been considered? If so, where can I read more about this? Is there any experiment one can do to prove we live in a non-zero genus world?
general-relativity differential-geometry topology specific-reference
asked 3 hours ago
More Anonymous
967
Imagine there is "cavity" in one's locally Lorentzian manifold (the manifold has non-zero genus). Have these kind of solutions in general relativity been considered? If so, where can I read more about this? Is there any experiment one can do to prove we live in a non-zero genus world?
general-relativity differential-geometry topology specific-reference
general-relativity differential-geometry topology specific-reference
asked 3 hours ago
More Anonymous
967
asked 3 hours ago
More Anonymous
967
asked 3 hours ago
More Anonymous
967
asked 3 hours ago
More Anonymous
967
asked 3 hours ago
More Anonymous
967
967
add a comment |Â
add a comment |Â
1 Answer
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Any manifold, regardless of its properties, is a solution to the Einstein field equations for some stress-energy tensor $T_munu$, providing it admits a metric, $g_munu$.
In theory, you are free to pick your favourite algebraic manifold of genus $g$, compute the metric and find its corresponding stress-energy tensor.
However, if we restrict the stress-energy, say by requiring that $T_munu$ be infinitely differentiable, then we also restrict the space of allowable metrics. Likewise, if we demand the manifold be globally hyperbolic, which is a frequent requirement due to the implication on causality, then we also restrict possible solutions.
Based on a cursory search, I cannot find any classification of genus $g$ spacetime solutions yet. However, in the paper here it is written that,
In the current paper we study that equation on closed 2-dimensional
surfaces that have genus $>0$. We derive all the solutions assuming
the embeddability in 4-dimensional spacetime that satisfies the vacuum
Einstein equations...
It seems we can have sub-manifolds in spacetime which do have non-zero genus, and are sensible solutions to the vacuum Einstein field equations, meaning at least $T_munu$ is physically sensible, being zero.
answered 2 hours ago
JamalS
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Any manifold, regardless of its properties, is a solution to the Einstein field equations for some stress-energy tensor $T_munu$, providing it admits a metric, $g_munu$.
In theory, you are free to pick your favourite algebraic manifold of genus $g$, compute the metric and find its corresponding stress-energy tensor.
However, if we restrict the stress-energy, say by requiring that $T_munu$ be infinitely differentiable, then we also restrict the space of allowable metrics. Likewise, if we demand the manifold be globally hyperbolic, which is a frequent requirement due to the implication on causality, then we also restrict possible solutions.
Based on a cursory search, I cannot find any classification of genus $g$ spacetime solutions yet. However, in the paper here it is written that,
In the current paper we study that equation on closed 2-dimensional
surfaces that have genus $>0$. We derive all the solutions assuming
the embeddability in 4-dimensional spacetime that satisfies the vacuum
Einstein equations...
It seems we can have sub-manifolds in spacetime which do have non-zero genus, and are sensible solutions to the vacuum Einstein field equations, meaning at least $T_munu$ is physically sensible, being zero.
answered 2 hours ago
JamalS
14k52983
add a comment |Â
up vote
4
down vote
Any manifold, regardless of its properties, is a solution to the Einstein field equations for some stress-energy tensor $T_munu$, providing it admits a metric, $g_munu$.
In theory, you are free to pick your favourite algebraic manifold of genus $g$, compute the metric and find its corresponding stress-energy tensor.
However, if we restrict the stress-energy, say by requiring that $T_munu$ be infinitely differentiable, then we also restrict the space of allowable metrics. Likewise, if we demand the manifold be globally hyperbolic, which is a frequent requirement due to the implication on causality, then we also restrict possible solutions.
Based on a cursory search, I cannot find any classification of genus $g$ spacetime solutions yet. However, in the paper here it is written that,
In the current paper we study that equation on closed 2-dimensional
surfaces that have genus $>0$. We derive all the solutions assuming
the embeddability in 4-dimensional spacetime that satisfies the vacuum
Einstein equations...
It seems we can have sub-manifolds in spacetime which do have non-zero genus, and are sensible solutions to the vacuum Einstein field equations, meaning at least $T_munu$ is physically sensible, being zero.
answered 2 hours ago
JamalS
14k52983
add a comment |Â
up vote
4
down vote
up vote
4
down vote
Any manifold, regardless of its properties, is a solution to the Einstein field equations for some stress-energy tensor $T_munu$, providing it admits a metric, $g_munu$.
In theory, you are free to pick your favourite algebraic manifold of genus $g$, compute the metric and find its corresponding stress-energy tensor.
However, if we restrict the stress-energy, say by requiring that $T_munu$ be infinitely differentiable, then we also restrict the space of allowable metrics. Likewise, if we demand the manifold be globally hyperbolic, which is a frequent requirement due to the implication on causality, then we also restrict possible solutions.
Based on a cursory search, I cannot find any classification of genus $g$ spacetime solutions yet. However, in the paper here it is written that,
In the current paper we study that equation on closed 2-dimensional
surfaces that have genus $>0$. We derive all the solutions assuming
the embeddability in 4-dimensional spacetime that satisfies the vacuum
Einstein equations...
It seems we can have sub-manifolds in spacetime which do have non-zero genus, and are sensible solutions to the vacuum Einstein field equations, meaning at least $T_munu$ is physically sensible, being zero.
answered 2 hours ago
JamalS
14k52983
Any manifold, regardless of its properties, is a solution to the Einstein field equations for some stress-energy tensor $T_munu$, providing it admits a metric, $g_munu$.
In theory, you are free to pick your favourite algebraic manifold of genus $g$, compute the metric and find its corresponding stress-energy tensor.
However, if we restrict the stress-energy, say by requiring that $T_munu$ be infinitely differentiable, then we also restrict the space of allowable metrics. Likewise, if we demand the manifold be globally hyperbolic, which is a frequent requirement due to the implication on causality, then we also restrict possible solutions.
Based on a cursory search, I cannot find any classification of genus $g$ spacetime solutions yet. However, in the paper here it is written that,
In the current paper we study that equation on closed 2-dimensional
surfaces that have genus $>0$. We derive all the solutions assuming
the embeddability in 4-dimensional spacetime that satisfies the vacuum
Einstein equations...
It seems we can have sub-manifolds in spacetime which do have non-zero genus, and are sensible solutions to the vacuum Einstein field equations, meaning at least $T_munu$ is physically sensible, being zero.
answered 2 hours ago
JamalS
14k52983
answered 2 hours ago
JamalS
14k52983
answered 2 hours ago
JamalS
14k52983
answered 2 hours ago
JamalS
14k52983
14k52983
add a comment |Â
add a comment |Â
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