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Why isn't an infinite, flat, nonexpanding universe a solution to Einstein's equation?
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In Newtonian gravity, an infinite volume filled with a uniform distribution of mass would be in perfect equilibrium. At every point, the gravitational forces contributed by masses in one direction would be exactly counterbalanced by those in the opposite direction.
But when Einstein tried to apply General Relativity to possible cosmologies, he found it necessary to include the cosmological constant in order to get a static universe.
In qualitative terms, it seems to me that the gravitational stresses that the masses would impose on the spacetime should all cancel out, and likewise, that the resulting flat spacetime should have no effect on the motion of the masses.
However, the math of the situation is beyond my current skills, so I'm asking how it produces the nonequilibrium condition?
(I realize that such an equilibrium solution might not be stable, and that there are many other very good reasons to believe in an expanding universe, so I'm not trying to promote any alternative theories. I'm just curious about this particular point. )
general-relativity cosmology
asked 2 hours ago
D. Halsey
39729
 |Â
show 3 more comments
up vote
6
down vote
favorite
In Newtonian gravity, an infinite volume filled with a uniform distribution of mass would be in perfect equilibrium. At every point, the gravitational forces contributed by masses in one direction would be exactly counterbalanced by those in the opposite direction.
But when Einstein tried to apply General Relativity to possible cosmologies, he found it necessary to include the cosmological constant in order to get a static universe.
In qualitative terms, it seems to me that the gravitational stresses that the masses would impose on the spacetime should all cancel out, and likewise, that the resulting flat spacetime should have no effect on the motion of the masses.
However, the math of the situation is beyond my current skills, so I'm asking how it produces the nonequilibrium condition?
(I realize that such an equilibrium solution might not be stable, and that there are many other very good reasons to believe in an expanding universe, so I'm not trying to promote any alternative theories. I'm just curious about this particular point. )
general-relativity cosmology
asked 2 hours ago
D. Halsey
39729
2
Infinite, flat and non-expanding metric is surely a solution to Einstein's equations: Minkowski metric $eta_mu nu$
– Avantgarde
1 hour ago
@Avantgarde: Then why didn't Einstein advocate that as his cosmology? Did he consider only finite universes?
– D. Halsey
1 hour ago
@Avantgarde: Is it still considered Minkowski space when the mass distribution is included?
– D. Halsey
1 hour ago
2
@Avantgarde: The OP is asking about a cosmology with a uniform mass distribution. Minkowski space isn't a solution to the Einstein field equations when the stress-energy tensor is nonzero.
– Ben Crowell
1 hour ago
1
@peterh: Spatial flatness is different from flatness of spacetime.
– Ben Crowell
8 mins ago
 |Â
show 3 more comments
up vote
6
down vote
favorite
up vote
6
down vote
favorite
In Newtonian gravity, an infinite volume filled with a uniform distribution of mass would be in perfect equilibrium. At every point, the gravitational forces contributed by masses in one direction would be exactly counterbalanced by those in the opposite direction.
But when Einstein tried to apply General Relativity to possible cosmologies, he found it necessary to include the cosmological constant in order to get a static universe.
In qualitative terms, it seems to me that the gravitational stresses that the masses would impose on the spacetime should all cancel out, and likewise, that the resulting flat spacetime should have no effect on the motion of the masses.
However, the math of the situation is beyond my current skills, so I'm asking how it produces the nonequilibrium condition?
(I realize that such an equilibrium solution might not be stable, and that there are many other very good reasons to believe in an expanding universe, so I'm not trying to promote any alternative theories. I'm just curious about this particular point. )
general-relativity cosmology
asked 2 hours ago
D. Halsey
39729
In Newtonian gravity, an infinite volume filled with a uniform distribution of mass would be in perfect equilibrium. At every point, the gravitational forces contributed by masses in one direction would be exactly counterbalanced by those in the opposite direction.
But when Einstein tried to apply General Relativity to possible cosmologies, he found it necessary to include the cosmological constant in order to get a static universe.
In qualitative terms, it seems to me that the gravitational stresses that the masses would impose on the spacetime should all cancel out, and likewise, that the resulting flat spacetime should have no effect on the motion of the masses.
However, the math of the situation is beyond my current skills, so I'm asking how it produces the nonequilibrium condition?
(I realize that such an equilibrium solution might not be stable, and that there are many other very good reasons to believe in an expanding universe, so I'm not trying to promote any alternative theories. I'm just curious about this particular point. )
general-relativity cosmology
general-relativity cosmology
asked 2 hours ago
D. Halsey
39729
asked 2 hours ago
D. Halsey
39729
asked 2 hours ago
D. Halsey
39729
asked 2 hours ago
D. Halsey
39729
asked 2 hours ago
D. Halsey
39729
39729
2
Infinite, flat and non-expanding metric is surely a solution to Einstein's equations: Minkowski metric $eta_mu nu$
– Avantgarde
1 hour ago
@Avantgarde: Then why didn't Einstein advocate that as his cosmology? Did he consider only finite universes?
– D. Halsey
1 hour ago
@Avantgarde: Is it still considered Minkowski space when the mass distribution is included?
– D. Halsey
1 hour ago
2
@Avantgarde: The OP is asking about a cosmology with a uniform mass distribution. Minkowski space isn't a solution to the Einstein field equations when the stress-energy tensor is nonzero.
– Ben Crowell
1 hour ago
1
@peterh: Spatial flatness is different from flatness of spacetime.
– Ben Crowell
8 mins ago
 |Â
show 3 more comments
2
Infinite, flat and non-expanding metric is surely a solution to Einstein's equations: Minkowski metric $eta_mu nu$
– Avantgarde
1 hour ago
@Avantgarde: Then why didn't Einstein advocate that as his cosmology? Did he consider only finite universes?
– D. Halsey
1 hour ago
@Avantgarde: Is it still considered Minkowski space when the mass distribution is included?
– D. Halsey
1 hour ago
2
@Avantgarde: The OP is asking about a cosmology with a uniform mass distribution. Minkowski space isn't a solution to the Einstein field equations when the stress-energy tensor is nonzero.
– Ben Crowell
1 hour ago
1
@peterh: Spatial flatness is different from flatness of spacetime.
– Ben Crowell
8 mins ago
2
2
Infinite, flat and non-expanding metric is surely a solution to Einstein's equations: Minkowski metric $eta_mu nu$
– Avantgarde
1 hour ago
Infinite, flat and non-expanding metric is surely a solution to Einstein's equations: Minkowski metric $eta_mu nu$
– Avantgarde
1 hour ago
@Avantgarde: Then why didn't Einstein advocate that as his cosmology? Did he consider only finite universes?
– D. Halsey
1 hour ago
@Avantgarde: Then why didn't Einstein advocate that as his cosmology? Did he consider only finite universes?
– D. Halsey
1 hour ago
@Avantgarde: Is it still considered Minkowski space when the mass distribution is included?
– D. Halsey
1 hour ago
@Avantgarde: Is it still considered Minkowski space when the mass distribution is included?
– D. Halsey
1 hour ago
2
2
@Avantgarde: The OP is asking about a cosmology with a uniform mass distribution. Minkowski space isn't a solution to the Einstein field equations when the stress-energy tensor is nonzero.
– Ben Crowell
1 hour ago
@Avantgarde: The OP is asking about a cosmology with a uniform mass distribution. Minkowski space isn't a solution to the Einstein field equations when the stress-energy tensor is nonzero.
– Ben Crowell
1 hour ago
1
1
@peterh: Spatial flatness is different from flatness of spacetime.
– Ben Crowell
8 mins ago
@peterh: Spatial flatness is different from flatness of spacetime.
– Ben Crowell
8 mins ago
 |Â
show 3 more comments
2 Answers
2
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oldest
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up vote
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Nice question!
Here's a possible statement of the logic in the newtonian case. (1) In newtonian mechanics, we assume that inertial reference frames exist (this is one popular modern way of restating Newton's first law), we assume that such frames are global, and we assume that we can always find such a frame by observing a test particle that is not acted on by any force. (2) In the newtonian homogeneous cosmology, we could assume that the force on a chosen test particle P can be found by some limiting process, and that the result is unique. (This is basically a bogus assumption, but I don't think that ends up being the issue here.) (3) Given that the result is unique, it must be zero by symmetry. (4) By assumptions 1 and 2, P defines an inertial frame, and by assumption 1, that frame can be extended to cover the entire universe. Therefore all other particles in the universe must have zero acceleration relative to P.
In general relativity, assumption 1 fails. Test particles P and Q can both be inertial (i.e., no nongravitational forces act on them), but it can be false that they are not accelerated relative to one another. For example, we can make an FRW cosmology in which, at some initial time, $dota=0$, but then it will have $ddotane0$ (in order to satisfy the Einstein field equations for a uniform dust). (In this situation, the Einstein field equations can be reduced to the Friedmann equations, one of which is $ddota/a=-(4pi/3)rho$.)
This shows that the newtonian argument (or at least one version of it) fails. It does not prove that there is no other semi-newtonian plausibility argument that explains why an initially static universe collapses. However, I'm not sure what criteria we would be able to agree on as to what constitutes an acceptable semi-newtonian plausibility argument. Some people have developed these semi-newtonian descriptions of cosmology at great length, but to me they appear to lack any logical foundations that would allow one to tell a correct argument from an incorrect one.
answered 40 mins ago
Ben Crowell
44.6k3146271
add a comment |Â
up vote
1
down vote
As other commenters on your question have pointed out, flat spacetime is a solution when there are no masses hanging around. It's only when the universe has some mass density that it stops being a viable solution. However Newtonian gravity doesn't really give viable predictions when dealing with an infinite universe of constant mass density. Consider the gravitational field at the red dot in the following figures:
Suppose that in the figure, space is filled with mass with some constant density. According to the shell theorem, a shell of constant mass density produces no gravitational field inside of it. Therefore, we can think of any spherical shell enclosing the red dot as having zero net gravitational effect. If we draw the spherical shells around the red dot as in fig a, then the gravitational field should be 0. If we draw them as in fig b, then it looks like the red dot is sitting on the surface of a planet, so the gravitational field should point to the right. Newtonian gravity predicts two different answers at the same time (in fact, infinitely many), which tells us that the theory breaks down here.
So General Relativity doesn't contradict Newton in this situation. It's merely the only theory that can make any predictions at all about cosmology in an infinite universe of uniform mass density.
answered 23 mins ago
Ricky Tensor
66616
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Nice question!
Here's a possible statement of the logic in the newtonian case. (1) In newtonian mechanics, we assume that inertial reference frames exist (this is one popular modern way of restating Newton's first law), we assume that such frames are global, and we assume that we can always find such a frame by observing a test particle that is not acted on by any force. (2) In the newtonian homogeneous cosmology, we could assume that the force on a chosen test particle P can be found by some limiting process, and that the result is unique. (This is basically a bogus assumption, but I don't think that ends up being the issue here.) (3) Given that the result is unique, it must be zero by symmetry. (4) By assumptions 1 and 2, P defines an inertial frame, and by assumption 1, that frame can be extended to cover the entire universe. Therefore all other particles in the universe must have zero acceleration relative to P.
In general relativity, assumption 1 fails. Test particles P and Q can both be inertial (i.e., no nongravitational forces act on them), but it can be false that they are not accelerated relative to one another. For example, we can make an FRW cosmology in which, at some initial time, $dota=0$, but then it will have $ddotane0$ (in order to satisfy the Einstein field equations for a uniform dust). (In this situation, the Einstein field equations can be reduced to the Friedmann equations, one of which is $ddota/a=-(4pi/3)rho$.)
This shows that the newtonian argument (or at least one version of it) fails. It does not prove that there is no other semi-newtonian plausibility argument that explains why an initially static universe collapses. However, I'm not sure what criteria we would be able to agree on as to what constitutes an acceptable semi-newtonian plausibility argument. Some people have developed these semi-newtonian descriptions of cosmology at great length, but to me they appear to lack any logical foundations that would allow one to tell a correct argument from an incorrect one.
answered 40 mins ago
Ben Crowell
44.6k3146271
add a comment |Â
up vote
4
down vote
Nice question!
Here's a possible statement of the logic in the newtonian case. (1) In newtonian mechanics, we assume that inertial reference frames exist (this is one popular modern way of restating Newton's first law), we assume that such frames are global, and we assume that we can always find such a frame by observing a test particle that is not acted on by any force. (2) In the newtonian homogeneous cosmology, we could assume that the force on a chosen test particle P can be found by some limiting process, and that the result is unique. (This is basically a bogus assumption, but I don't think that ends up being the issue here.) (3) Given that the result is unique, it must be zero by symmetry. (4) By assumptions 1 and 2, P defines an inertial frame, and by assumption 1, that frame can be extended to cover the entire universe. Therefore all other particles in the universe must have zero acceleration relative to P.
In general relativity, assumption 1 fails. Test particles P and Q can both be inertial (i.e., no nongravitational forces act on them), but it can be false that they are not accelerated relative to one another. For example, we can make an FRW cosmology in which, at some initial time, $dota=0$, but then it will have $ddotane0$ (in order to satisfy the Einstein field equations for a uniform dust). (In this situation, the Einstein field equations can be reduced to the Friedmann equations, one of which is $ddota/a=-(4pi/3)rho$.)
This shows that the newtonian argument (or at least one version of it) fails. It does not prove that there is no other semi-newtonian plausibility argument that explains why an initially static universe collapses. However, I'm not sure what criteria we would be able to agree on as to what constitutes an acceptable semi-newtonian plausibility argument. Some people have developed these semi-newtonian descriptions of cosmology at great length, but to me they appear to lack any logical foundations that would allow one to tell a correct argument from an incorrect one.
answered 40 mins ago
Ben Crowell
44.6k3146271
add a comment |Â
up vote
4
down vote
up vote
4
down vote
Nice question!
Here's a possible statement of the logic in the newtonian case. (1) In newtonian mechanics, we assume that inertial reference frames exist (this is one popular modern way of restating Newton's first law), we assume that such frames are global, and we assume that we can always find such a frame by observing a test particle that is not acted on by any force. (2) In the newtonian homogeneous cosmology, we could assume that the force on a chosen test particle P can be found by some limiting process, and that the result is unique. (This is basically a bogus assumption, but I don't think that ends up being the issue here.) (3) Given that the result is unique, it must be zero by symmetry. (4) By assumptions 1 and 2, P defines an inertial frame, and by assumption 1, that frame can be extended to cover the entire universe. Therefore all other particles in the universe must have zero acceleration relative to P.
In general relativity, assumption 1 fails. Test particles P and Q can both be inertial (i.e., no nongravitational forces act on them), but it can be false that they are not accelerated relative to one another. For example, we can make an FRW cosmology in which, at some initial time, $dota=0$, but then it will have $ddotane0$ (in order to satisfy the Einstein field equations for a uniform dust). (In this situation, the Einstein field equations can be reduced to the Friedmann equations, one of which is $ddota/a=-(4pi/3)rho$.)
This shows that the newtonian argument (or at least one version of it) fails. It does not prove that there is no other semi-newtonian plausibility argument that explains why an initially static universe collapses. However, I'm not sure what criteria we would be able to agree on as to what constitutes an acceptable semi-newtonian plausibility argument. Some people have developed these semi-newtonian descriptions of cosmology at great length, but to me they appear to lack any logical foundations that would allow one to tell a correct argument from an incorrect one.
answered 40 mins ago
Ben Crowell
44.6k3146271
Nice question!
Here's a possible statement of the logic in the newtonian case. (1) In newtonian mechanics, we assume that inertial reference frames exist (this is one popular modern way of restating Newton's first law), we assume that such frames are global, and we assume that we can always find such a frame by observing a test particle that is not acted on by any force. (2) In the newtonian homogeneous cosmology, we could assume that the force on a chosen test particle P can be found by some limiting process, and that the result is unique. (This is basically a bogus assumption, but I don't think that ends up being the issue here.) (3) Given that the result is unique, it must be zero by symmetry. (4) By assumptions 1 and 2, P defines an inertial frame, and by assumption 1, that frame can be extended to cover the entire universe. Therefore all other particles in the universe must have zero acceleration relative to P.
In general relativity, assumption 1 fails. Test particles P and Q can both be inertial (i.e., no nongravitational forces act on them), but it can be false that they are not accelerated relative to one another. For example, we can make an FRW cosmology in which, at some initial time, $dota=0$, but then it will have $ddotane0$ (in order to satisfy the Einstein field equations for a uniform dust). (In this situation, the Einstein field equations can be reduced to the Friedmann equations, one of which is $ddota/a=-(4pi/3)rho$.)
This shows that the newtonian argument (or at least one version of it) fails. It does not prove that there is no other semi-newtonian plausibility argument that explains why an initially static universe collapses. However, I'm not sure what criteria we would be able to agree on as to what constitutes an acceptable semi-newtonian plausibility argument. Some people have developed these semi-newtonian descriptions of cosmology at great length, but to me they appear to lack any logical foundations that would allow one to tell a correct argument from an incorrect one.
answered 40 mins ago
Ben Crowell
44.6k3146271
edited 13 mins ago
answered 40 mins ago
Ben Crowell
44.6k3146271
answered 40 mins ago
Ben Crowell
44.6k3146271
answered 40 mins ago
Ben Crowell
44.6k3146271
44.6k3146271
add a comment |Â
add a comment |Â
up vote
1
down vote
As other commenters on your question have pointed out, flat spacetime is a solution when there are no masses hanging around. It's only when the universe has some mass density that it stops being a viable solution. However Newtonian gravity doesn't really give viable predictions when dealing with an infinite universe of constant mass density. Consider the gravitational field at the red dot in the following figures:
Suppose that in the figure, space is filled with mass with some constant density. According to the shell theorem, a shell of constant mass density produces no gravitational field inside of it. Therefore, we can think of any spherical shell enclosing the red dot as having zero net gravitational effect. If we draw the spherical shells around the red dot as in fig a, then the gravitational field should be 0. If we draw them as in fig b, then it looks like the red dot is sitting on the surface of a planet, so the gravitational field should point to the right. Newtonian gravity predicts two different answers at the same time (in fact, infinitely many), which tells us that the theory breaks down here.
So General Relativity doesn't contradict Newton in this situation. It's merely the only theory that can make any predictions at all about cosmology in an infinite universe of uniform mass density.
answered 23 mins ago
Ricky Tensor
66616
add a comment |Â
up vote
1
down vote
As other commenters on your question have pointed out, flat spacetime is a solution when there are no masses hanging around. It's only when the universe has some mass density that it stops being a viable solution. However Newtonian gravity doesn't really give viable predictions when dealing with an infinite universe of constant mass density. Consider the gravitational field at the red dot in the following figures:
Suppose that in the figure, space is filled with mass with some constant density. According to the shell theorem, a shell of constant mass density produces no gravitational field inside of it. Therefore, we can think of any spherical shell enclosing the red dot as having zero net gravitational effect. If we draw the spherical shells around the red dot as in fig a, then the gravitational field should be 0. If we draw them as in fig b, then it looks like the red dot is sitting on the surface of a planet, so the gravitational field should point to the right. Newtonian gravity predicts two different answers at the same time (in fact, infinitely many), which tells us that the theory breaks down here.
So General Relativity doesn't contradict Newton in this situation. It's merely the only theory that can make any predictions at all about cosmology in an infinite universe of uniform mass density.
answered 23 mins ago
Ricky Tensor
66616
add a comment |Â
up vote
1
down vote
up vote
1
down vote
As other commenters on your question have pointed out, flat spacetime is a solution when there are no masses hanging around. It's only when the universe has some mass density that it stops being a viable solution. However Newtonian gravity doesn't really give viable predictions when dealing with an infinite universe of constant mass density. Consider the gravitational field at the red dot in the following figures:
Suppose that in the figure, space is filled with mass with some constant density. According to the shell theorem, a shell of constant mass density produces no gravitational field inside of it. Therefore, we can think of any spherical shell enclosing the red dot as having zero net gravitational effect. If we draw the spherical shells around the red dot as in fig a, then the gravitational field should be 0. If we draw them as in fig b, then it looks like the red dot is sitting on the surface of a planet, so the gravitational field should point to the right. Newtonian gravity predicts two different answers at the same time (in fact, infinitely many), which tells us that the theory breaks down here.
So General Relativity doesn't contradict Newton in this situation. It's merely the only theory that can make any predictions at all about cosmology in an infinite universe of uniform mass density.
answered 23 mins ago
Ricky Tensor
66616
As other commenters on your question have pointed out, flat spacetime is a solution when there are no masses hanging around. It's only when the universe has some mass density that it stops being a viable solution. However Newtonian gravity doesn't really give viable predictions when dealing with an infinite universe of constant mass density. Consider the gravitational field at the red dot in the following figures:
Suppose that in the figure, space is filled with mass with some constant density. According to the shell theorem, a shell of constant mass density produces no gravitational field inside of it. Therefore, we can think of any spherical shell enclosing the red dot as having zero net gravitational effect. If we draw the spherical shells around the red dot as in fig a, then the gravitational field should be 0. If we draw them as in fig b, then it looks like the red dot is sitting on the surface of a planet, so the gravitational field should point to the right. Newtonian gravity predicts two different answers at the same time (in fact, infinitely many), which tells us that the theory breaks down here.
So General Relativity doesn't contradict Newton in this situation. It's merely the only theory that can make any predictions at all about cosmology in an infinite universe of uniform mass density.
answered 23 mins ago
Ricky Tensor
66616
answered 23 mins ago
Ricky Tensor
66616
answered 23 mins ago
Ricky Tensor
66616
answered 23 mins ago
Ricky Tensor
66616
66616
add a comment |Â
add a comment |Â
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2
Infinite, flat and non-expanding metric is surely a solution to Einstein's equations: Minkowski metric $eta_mu nu$
– Avantgarde
1 hour ago
@Avantgarde: Then why didn't Einstein advocate that as his cosmology? Did he consider only finite universes?
– D. Halsey
1 hour ago
@Avantgarde: Is it still considered Minkowski space when the mass distribution is included?
– D. Halsey
1 hour ago
2
@Avantgarde: The OP is asking about a cosmology with a uniform mass distribution. Minkowski space isn't a solution to the Einstein field equations when the stress-energy tensor is nonzero.
– Ben Crowell
1 hour ago
1
@peterh: Spatial flatness is different from flatness of spacetime.
– Ben Crowell
8 mins ago