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#1
Oct1206, 05:11 AM

P: n/a

What properties of the big bang singularity can be inferred from the
PenroseHawking theorem? What properties of the big bang singularity can be inferred from modern physics as a whole? By properties I mean things like charge, mass, and spin. Thanks, Michael S. 


#2
Oct1206, 05:11 AM

P: n/a

On Tue, 25 Oct 2005, Michael S. <thesparr0w1@yahoo.com> wrote:
> What properties of the big bang singularity can be inferred from the > PenroseHawking theorem? What properties of the big bang singularity > can be inferred from modern physics as a whole? By properties I mean > things like charge, mass, and spin. The problem word here is "properties". Charge, mass, and angular momentum are properties of "physical objects" like stars. In general relativity, it is not hard to define them for isolated objects (most recent textbooks explain this), but in more general situations this can get rather tricky. (Advanced students can search on the keyphrase "quasilocal mass", among other relevant topics.) The external field of a black hole (outside some compact region) will be similar or identical to that of an appropriately chosen model of an uncollapsed object like a star, so the elementary "isolated object" definitions do apply to isolated black holes. That's good enough for astrophysicists and cosmologists to talk about a "twohundred million solar mass black hole" without tying themselves into knots over fine points. Or indeed, to call such a thing an "object" on a par with objects such as stars. In a field theory, a "singularity" is basically a locus (location in a manifold) where our mathematical description of a model of some physical scenario breaks down. Sometimes our description breaks down but changing to another equally valid description (in this context, this probably means changing to another "coordinate chart") removes the problem, so these are called "removable singularities". But sometimes this doesn't work: the theory predicts some kind of behavior in the field which it just cannot handle no matter what description we attempt to employ. These are "irremovable singularities". In "metric theories" like gtr, we may encounter irremovable singularities in the curvature tensor of some spacetime. The point is that such singularities are properties of spacetime itself, not of any "physical object". To get a better sense of why the distinction is so important, we should take a closer look at singularities. Unfortunately, an enlightening general discussion is hard to come by, because curvature singularities are much harder to study in Lorentzian (indefinite signature) than Riemannian (positive definite signature) manifolds. There have been some attempts to classify them, but it seems fair to describe these attempts as preliminary and phenomenological. Commonly encountered types of curvature singularities include the following: 1. Scalar vs. nonscalar singularities: polynomial invariants like the Kretschmann scalar R_(abcd) R^(abcd) (this is the bestknown example of a degree two or quadratic invariant) may or may not blow up. Schwarzschild and Kerr vacuum: yes; ppwaves: no. Indeed, ppwaves always have identically vanishing polynomial invariants, but there are examples which nonetheless possess irremovable nonscalar curvature singularities). 2. Timelike, spacelike, null: some singular locuses in some spacetimes can be described, roughly speaking, as the limit of a family of timelike, spacelike, or null hyperslices. (This idea is most often applied in the context of a Penrose diagram used to visualize the conformal structure of a given spacetime.) For example, in the FRW models, we have a spacelike singularity. This can get tricky since unexpected things can happen: most Kerr vacuum solutions have timelike curvature singularities (inside horizons), but the Schwarzschild special case has a spacelike singularity. 3. Strong vs. weak singularities: sometimes two ideal observers whose worldlines are two different timelike curves in the same spacetime can have drastically different physical experiences as they approach the same singular locus. In weak singularities, some observers can apparently survive the encounter because roughly speaking they experience tidal forces which do indeed blow up, but so briefly that the net effect on their bodies is negligible. This can be seen in phenomena where the expansion tensor of some congruence remains finite while the curvature tensor very rapidly blows up. 4. The weakest or mildest type of irremovable singularities closely resemble certain kinds of coordinate singularities. For example, "cone singularities" occur in certain cosmic string "solutions" (look at a cone and think of the apex as a zerodimensional "reduction" of the onedimensional symmetry axis in a cylindrical coordinate chart on a manifold which is locally isometric to ordinary euclidean three dimensional space off the axis), and are also associated with unphysical "massless struts" which occur in certain axisymmetric static vacuum solutions in gtr. (See also "orbifolds" in pure mathematics.) Another example are "fold singularities" in certain colliding plane wave solutions, including the wellknown PenroseKhan CPW solution. Going back to black holes: it is true that historically one place where singularities first arose in classical field theories was in the notion of a "point mass", and at least at first sight, black hole solutions in general relativity are analogous to such point masses. However, gtr is mathematically and conceptually significantly more challenging than Maxwell's theory of EM (for example), and these analogies, while valid as far as they go, are potentially misleading. One final comment: for the topic at hand, the key difference between black holes and stars is probably that stars have a welldefined surface. (Not real stars, of course, which have a corona, stellar wind, and all that, but highly idealized stars treated as a ball of perfect fluid held together by its own gravity.) This surface is not really analogous to a black hole horizon, and it turns out that one cannot obtain, in gtr, a family of stellar models with surfaces which approach say a Schwarzschild hole, with the surfaces approaching the horizon. (Keyword: Buchdahl's theorem.) "T. Essel" 


#3
Oct1206, 05:11 AM

P: n/a

On Tue, 25 Oct 2005, Michael S. <thesparr0w1@yahoo.com> wrote:
> What properties of the big bang singularity can be inferred from the > PenroseHawking theorem? What properties of the big bang singularity > can be inferred from modern physics as a whole? By properties I mean > things like charge, mass, and spin. The problem word here is "properties". Charge, mass, and angular momentum are properties of "physical objects" like stars. In general relativity, it is not hard to define them for isolated objects (most recent textbooks explain this), but in more general situations this can get rather tricky. (Advanced students can search on the keyphrase "quasilocal mass", among other relevant topics.) The external field of a black hole (outside some compact region) will be similar or identical to that of an appropriately chosen model of an uncollapsed object like a star, so the elementary "isolated object" definitions do apply to isolated black holes. That's good enough for astrophysicists and cosmologists to talk about a "twohundred million solar mass black hole" without tying themselves into knots over fine points. Or indeed, to call such a thing an "object" on a par with objects such as stars. In a field theory, a "singularity" is basically a locus (location in a manifold) where our mathematical description of a model of some physical scenario breaks down. Sometimes our description breaks down but changing to another equally valid description (in this context, this probably means changing to another "coordinate chart") removes the problem, so these are called "removable singularities". But sometimes this doesn't work: the theory predicts some kind of behavior in the field which it just cannot handle no matter what description we attempt to employ. These are "irremovable singularities". In "metric theories" like gtr, we may encounter irremovable singularities in the curvature tensor of some spacetime. The point is that such singularities are properties of spacetime itself, not of any "physical object". To get a better sense of why the distinction is so important, we should take a closer look at singularities. Unfortunately, an enlightening general discussion is hard to come by, because curvature singularities are much harder to study in Lorentzian (indefinite signature) than Riemannian (positive definite signature) manifolds. There have been some attempts to classify them, but it seems fair to describe these attempts as preliminary and phenomenological. Commonly encountered types of curvature singularities include the following: 1. Scalar vs. nonscalar singularities: polynomial invariants like the Kretschmann scalar R_(abcd) R^(abcd) (this is the bestknown example of a degree two or quadratic invariant) may or may not blow up. Schwarzschild and Kerr vacuum: yes; ppwaves: no. Indeed, ppwaves always have identically vanishing polynomial invariants, but there are examples which nonetheless possess irremovable nonscalar curvature singularities). 2. Timelike, spacelike, null: some singular locuses in some spacetimes can be described, roughly speaking, as the limit of a family of timelike, spacelike, or null hyperslices. (This idea is most often applied in the context of a Penrose diagram used to visualize the conformal structure of a given spacetime.) For example, in the FRW models, we have a spacelike singularity. This can get tricky since unexpected things can happen: most Kerr vacuum solutions have timelike curvature singularities (inside horizons), but the Schwarzschild special case has a spacelike singularity. 3. Strong vs. weak singularities: sometimes two ideal observers whose worldlines are two different timelike curves in the same spacetime can have drastically different physical experiences as they approach the same singular locus. In weak singularities, some observers can apparently survive the encounter because roughly speaking they experience tidal forces which do indeed blow up, but so briefly that the net effect on their bodies is negligible. This can be seen in phenomena where the expansion tensor of some congruence remains finite while the curvature tensor very rapidly blows up. 4. The weakest or mildest type of irremovable singularities closely resemble certain kinds of coordinate singularities. For example, "cone singularities" occur in certain cosmic string "solutions" (look at a cone and think of the apex as a zerodimensional "reduction" of the onedimensional symmetry axis in a cylindrical coordinate chart on a manifold which is locally isometric to ordinary euclidean three dimensional space off the axis), and are also associated with unphysical "massless struts" which occur in certain axisymmetric static vacuum solutions in gtr. (See also "orbifolds" in pure mathematics.) Another example are "fold singularities" in certain colliding plane wave solutions, including the wellknown PenroseKhan CPW solution. Going back to black holes: it is true that historically one place where singularities first arose in classical field theories was in the notion of a "point mass", and at least at first sight, black hole solutions in general relativity are analogous to such point masses. However, gtr is mathematically and conceptually significantly more challenging than Maxwell's theory of EM (for example), and these analogies, while valid as far as they go, are potentially misleading. One final comment: for the topic at hand, the key difference between black holes and stars is probably that stars have a welldefined surface. (Not real stars, of course, which have a corona, stellar wind, and all that, but highly idealized stars treated as a ball of perfect fluid held together by its own gravity.) This surface is not really analogous to a black hole horizon, and it turns out that one cannot obtain, in gtr, a family of stellar models with surfaces which approach say a Schwarzschild hole, with the surfaces approaching the horizon. (Keyword: Buchdahl's theorem.) "T. Essel" 


#4
Oct1206, 05:11 AM

P: n/a

Big Bang singularity
On Tue, 25 Oct 2005, Michael S. <thesparr0w1@yahoo.com> wrote:
> What properties of the big bang singularity can be inferred from the > PenroseHawking theorem? What properties of the big bang singularity > can be inferred from modern physics as a whole? By properties I mean > things like charge, mass, and spin. The problem word here is "properties". Charge, mass, and angular momentum are properties of "physical objects" like stars. In general relativity, it is not hard to define them for isolated objects (most recent textbooks explain this), but in more general situations this can get rather tricky. (Advanced students can search on the keyphrase "quasilocal mass", among other relevant topics.) The external field of a black hole (outside some compact region) will be similar or identical to that of an appropriately chosen model of an uncollapsed object like a star, so the elementary "isolated object" definitions do apply to isolated black holes. That's good enough for astrophysicists and cosmologists to talk about a "twohundred million solar mass black hole" without tying themselves into knots over fine points. Or indeed, to call such a thing an "object" on a par with objects such as stars. In a field theory, a "singularity" is basically a locus (location in a manifold) where our mathematical description of a model of some physical scenario breaks down. Sometimes our description breaks down but changing to another equally valid description (in this context, this probably means changing to another "coordinate chart") removes the problem, so these are called "removable singularities". But sometimes this doesn't work: the theory predicts some kind of behavior in the field which it just cannot handle no matter what description we attempt to employ. These are "irremovable singularities". In "metric theories" like gtr, we may encounter irremovable singularities in the curvature tensor of some spacetime. The point is that such singularities are properties of spacetime itself, not of any "physical object". To get a better sense of why the distinction is so important, we should take a closer look at singularities. Unfortunately, an enlightening general discussion is hard to come by, because curvature singularities are much harder to study in Lorentzian (indefinite signature) than Riemannian (positive definite signature) manifolds. There have been some attempts to classify them, but it seems fair to describe these attempts as preliminary and phenomenological. Commonly encountered types of curvature singularities include the following: 1. Scalar vs. nonscalar singularities: polynomial invariants like the Kretschmann scalar R_(abcd) R^(abcd) (this is the bestknown example of a degree two or quadratic invariant) may or may not blow up. Schwarzschild and Kerr vacuum: yes; ppwaves: no. Indeed, ppwaves always have identically vanishing polynomial invariants, but there are examples which nonetheless possess irremovable nonscalar curvature singularities). 2. Timelike, spacelike, null: some singular locuses in some spacetimes can be described, roughly speaking, as the limit of a family of timelike, spacelike, or null hyperslices. (This idea is most often applied in the context of a Penrose diagram used to visualize the conformal structure of a given spacetime.) For example, in the FRW models, we have a spacelike singularity. This can get tricky since unexpected things can happen: most Kerr vacuum solutions have timelike curvature singularities (inside horizons), but the Schwarzschild special case has a spacelike singularity. 3. Strong vs. weak singularities: sometimes two ideal observers whose worldlines are two different timelike curves in the same spacetime can have drastically different physical experiences as they approach the same singular locus. In weak singularities, some observers can apparently survive the encounter because roughly speaking they experience tidal forces which do indeed blow up, but so briefly that the net effect on their bodies is negligible. This can be seen in phenomena where the expansion tensor of some congruence remains finite while the curvature tensor very rapidly blows up. 4. The weakest or mildest type of irremovable singularities closely resemble certain kinds of coordinate singularities. For example, "cone singularities" occur in certain cosmic string "solutions" (look at a cone and think of the apex as a zerodimensional "reduction" of the onedimensional symmetry axis in a cylindrical coordinate chart on a manifold which is locally isometric to ordinary euclidean three dimensional space off the axis), and are also associated with unphysical "massless struts" which occur in certain axisymmetric static vacuum solutions in gtr. (See also "orbifolds" in pure mathematics.) Another example are "fold singularities" in certain colliding plane wave solutions, including the wellknown PenroseKhan CPW solution. Going back to black holes: it is true that historically one place where singularities first arose in classical field theories was in the notion of a "point mass", and at least at first sight, black hole solutions in general relativity are analogous to such point masses. However, gtr is mathematically and conceptually significantly more challenging than Maxwell's theory of EM (for example), and these analogies, while valid as far as they go, are potentially misleading. One final comment: for the topic at hand, the key difference between black holes and stars is probably that stars have a welldefined surface. (Not real stars, of course, which have a corona, stellar wind, and all that, but highly idealized stars treated as a ball of perfect fluid held together by its own gravity.) This surface is not really analogous to a black hole horizon, and it turns out that one cannot obtain, in gtr, a family of stellar models with surfaces which approach say a Schwarzschild hole, with the surfaces approaching the horizon. (Keyword: Buchdahl's theorem.) "T. Essel" 


#5
Oct1206, 05:11 AM

P: n/a

On Tue, 25 Oct 2005, Michael S. <thesparr0w1@yahoo.com> wrote:
> What properties of the big bang singularity can be inferred from the > PenroseHawking theorem? What properties of the big bang singularity > can be inferred from modern physics as a whole? By properties I mean > things like charge, mass, and spin. The problem word here is "properties". Charge, mass, and angular momentum are properties of "physical objects" like stars. In general relativity, it is not hard to define them for isolated objects (most recent textbooks explain this), but in more general situations this can get rather tricky. (Advanced students can search on the keyphrase "quasilocal mass", among other relevant topics.) The external field of a black hole (outside some compact region) will be similar or identical to that of an appropriately chosen model of an uncollapsed object like a star, so the elementary "isolated object" definitions do apply to isolated black holes. That's good enough for astrophysicists and cosmologists to talk about a "twohundred million solar mass black hole" without tying themselves into knots over fine points. Or indeed, to call such a thing an "object" on a par with objects such as stars. In a field theory, a "singularity" is basically a locus (location in a manifold) where our mathematical description of a model of some physical scenario breaks down. Sometimes our description breaks down but changing to another equally valid description (in this context, this probably means changing to another "coordinate chart") removes the problem, so these are called "removable singularities". But sometimes this doesn't work: the theory predicts some kind of behavior in the field which it just cannot handle no matter what description we attempt to employ. These are "irremovable singularities". In "metric theories" like gtr, we may encounter irremovable singularities in the curvature tensor of some spacetime. The point is that such singularities are properties of spacetime itself, not of any "physical object". To get a better sense of why the distinction is so important, we should take a closer look at singularities. Unfortunately, an enlightening general discussion is hard to come by, because curvature singularities are much harder to study in Lorentzian (indefinite signature) than Riemannian (positive definite signature) manifolds. There have been some attempts to classify them, but it seems fair to describe these attempts as preliminary and phenomenological. Commonly encountered types of curvature singularities include the following: 1. Scalar vs. nonscalar singularities: polynomial invariants like the Kretschmann scalar R_(abcd) R^(abcd) (this is the bestknown example of a degree two or quadratic invariant) may or may not blow up. Schwarzschild and Kerr vacuum: yes; ppwaves: no. Indeed, ppwaves always have identically vanishing polynomial invariants, but there are examples which nonetheless possess irremovable nonscalar curvature singularities). 2. Timelike, spacelike, null: some singular locuses in some spacetimes can be described, roughly speaking, as the limit of a family of timelike, spacelike, or null hyperslices. (This idea is most often applied in the context of a Penrose diagram used to visualize the conformal structure of a given spacetime.) For example, in the FRW models, we have a spacelike singularity. This can get tricky since unexpected things can happen: most Kerr vacuum solutions have timelike curvature singularities (inside horizons), but the Schwarzschild special case has a spacelike singularity. 3. Strong vs. weak singularities: sometimes two ideal observers whose worldlines are two different timelike curves in the same spacetime can have drastically different physical experiences as they approach the same singular locus. In weak singularities, some observers can apparently survive the encounter because roughly speaking they experience tidal forces which do indeed blow up, but so briefly that the net effect on their bodies is negligible. This can be seen in phenomena where the expansion tensor of some congruence remains finite while the curvature tensor very rapidly blows up. 4. The weakest or mildest type of irremovable singularities closely resemble certain kinds of coordinate singularities. For example, "cone singularities" occur in certain cosmic string "solutions" (look at a cone and think of the apex as a zerodimensional "reduction" of the onedimensional symmetry axis in a cylindrical coordinate chart on a manifold which is locally isometric to ordinary euclidean three dimensional space off the axis), and are also associated with unphysical "massless struts" which occur in certain axisymmetric static vacuum solutions in gtr. (See also "orbifolds" in pure mathematics.) Another example are "fold singularities" in certain colliding plane wave solutions, including the wellknown PenroseKhan CPW solution. Going back to black holes: it is true that historically one place where singularities first arose in classical field theories was in the notion of a "point mass", and at least at first sight, black hole solutions in general relativity are analogous to such point masses. However, gtr is mathematically and conceptually significantly more challenging than Maxwell's theory of EM (for example), and these analogies, while valid as far as they go, are potentially misleading. One final comment: for the topic at hand, the key difference between black holes and stars is probably that stars have a welldefined surface. (Not real stars, of course, which have a corona, stellar wind, and all that, but highly idealized stars treated as a ball of perfect fluid held together by its own gravity.) This surface is not really analogous to a black hole horizon, and it turns out that one cannot obtain, in gtr, a family of stellar models with surfaces which approach say a Schwarzschild hole, with the surfaces approaching the horizon. (Keyword: Buchdahl's theorem.) "T. Essel" 


#6
Oct1206, 05:11 AM

P: n/a

On Tue, 25 Oct 2005, Michael S. <thesparr0w1@yahoo.com> wrote:
> What properties of the big bang singularity can be inferred from the > PenroseHawking theorem? What properties of the big bang singularity > can be inferred from modern physics as a whole? By properties I mean > things like charge, mass, and spin. The problem word here is "properties". Charge, mass, and angular momentum are properties of "physical objects" like stars. In general relativity, it is not hard to define them for isolated objects (most recent textbooks explain this), but in more general situations this can get rather tricky. (Advanced students can search on the keyphrase "quasilocal mass", among other relevant topics.) The external field of a black hole (outside some compact region) will be similar or identical to that of an appropriately chosen model of an uncollapsed object like a star, so the elementary "isolated object" definitions do apply to isolated black holes. That's good enough for astrophysicists and cosmologists to talk about a "twohundred million solar mass black hole" without tying themselves into knots over fine points. Or indeed, to call such a thing an "object" on a par with objects such as stars. In a field theory, a "singularity" is basically a locus (location in a manifold) where our mathematical description of a model of some physical scenario breaks down. Sometimes our description breaks down but changing to another equally valid description (in this context, this probably means changing to another "coordinate chart") removes the problem, so these are called "removable singularities". But sometimes this doesn't work: the theory predicts some kind of behavior in the field which it just cannot handle no matter what description we attempt to employ. These are "irremovable singularities". In "metric theories" like gtr, we may encounter irremovable singularities in the curvature tensor of some spacetime. The point is that such singularities are properties of spacetime itself, not of any "physical object". To get a better sense of why the distinction is so important, we should take a closer look at singularities. Unfortunately, an enlightening general discussion is hard to come by, because curvature singularities are much harder to study in Lorentzian (indefinite signature) than Riemannian (positive definite signature) manifolds. There have been some attempts to classify them, but it seems fair to describe these attempts as preliminary and phenomenological. Commonly encountered types of curvature singularities include the following: 1. Scalar vs. nonscalar singularities: polynomial invariants like the Kretschmann scalar R_(abcd) R^(abcd) (this is the bestknown example of a degree two or quadratic invariant) may or may not blow up. Schwarzschild and Kerr vacuum: yes; ppwaves: no. Indeed, ppwaves always have identically vanishing polynomial invariants, but there are examples which nonetheless possess irremovable nonscalar curvature singularities). 2. Timelike, spacelike, null: some singular locuses in some spacetimes can be described, roughly speaking, as the limit of a family of timelike, spacelike, or null hyperslices. (This idea is most often applied in the context of a Penrose diagram used to visualize the conformal structure of a given spacetime.) For example, in the FRW models, we have a spacelike singularity. This can get tricky since unexpected things can happen: most Kerr vacuum solutions have timelike curvature singularities (inside horizons), but the Schwarzschild special case has a spacelike singularity. 3. Strong vs. weak singularities: sometimes two ideal observers whose worldlines are two different timelike curves in the same spacetime can have drastically different physical experiences as they approach the same singular locus. In weak singularities, some observers can apparently survive the encounter because roughly speaking they experience tidal forces which do indeed blow up, but so briefly that the net effect on their bodies is negligible. This can be seen in phenomena where the expansion tensor of some congruence remains finite while the curvature tensor very rapidly blows up. 4. The weakest or mildest type of irremovable singularities closely resemble certain kinds of coordinate singularities. For example, "cone singularities" occur in certain cosmic string "solutions" (look at a cone and think of the apex as a zerodimensional "reduction" of the onedimensional symmetry axis in a cylindrical coordinate chart on a manifold which is locally isometric to ordinary euclidean three dimensional space off the axis), and are also associated with unphysical "massless struts" which occur in certain axisymmetric static vacuum solutions in gtr. (See also "orbifolds" in pure mathematics.) Another example are "fold singularities" in certain colliding plane wave solutions, including the wellknown PenroseKhan CPW solution. Going back to black holes: it is true that historically one place where singularities first arose in classical field theories was in the notion of a "point mass", and at least at first sight, black hole solutions in general relativity are analogous to such point masses. However, gtr is mathematically and conceptually significantly more challenging than Maxwell's theory of EM (for example), and these analogies, while valid as far as they go, are potentially misleading. One final comment: for the topic at hand, the key difference between black holes and stars is probably that stars have a welldefined surface. (Not real stars, of course, which have a corona, stellar wind, and all that, but highly idealized stars treated as a ball of perfect fluid held together by its own gravity.) This surface is not really analogous to a black hole horizon, and it turns out that one cannot obtain, in gtr, a family of stellar models with surfaces which approach say a Schwarzschild hole, with the surfaces approaching the horizon. (Keyword: Buchdahl's theorem.) "T. Essel" 


#7
Oct1206, 05:11 AM

P: n/a

On Tue, 25 Oct 2005, Michael S. <thesparr0w1@yahoo.com> wrote:
> What properties of the big bang singularity can be inferred from the > PenroseHawking theorem? What properties of the big bang singularity > can be inferred from modern physics as a whole? By properties I mean > things like charge, mass, and spin. The problem word here is "properties". Charge, mass, and angular momentum are properties of "physical objects" like stars. In general relativity, it is not hard to define them for isolated objects (most recent textbooks explain this), but in more general situations this can get rather tricky. (Advanced students can search on the keyphrase "quasilocal mass", among other relevant topics.) The external field of a black hole (outside some compact region) will be similar or identical to that of an appropriately chosen model of an uncollapsed object like a star, so the elementary "isolated object" definitions do apply to isolated black holes. That's good enough for astrophysicists and cosmologists to talk about a "twohundred million solar mass black hole" without tying themselves into knots over fine points. Or indeed, to call such a thing an "object" on a par with objects such as stars. In a field theory, a "singularity" is basically a locus (location in a manifold) where our mathematical description of a model of some physical scenario breaks down. Sometimes our description breaks down but changing to another equally valid description (in this context, this probably means changing to another "coordinate chart") removes the problem, so these are called "removable singularities". But sometimes this doesn't work: the theory predicts some kind of behavior in the field which it just cannot handle no matter what description we attempt to employ. These are "irremovable singularities". In "metric theories" like gtr, we may encounter irremovable singularities in the curvature tensor of some spacetime. The point is that such singularities are properties of spacetime itself, not of any "physical object". To get a better sense of why the distinction is so important, we should take a closer look at singularities. Unfortunately, an enlightening general discussion is hard to come by, because curvature singularities are much harder to study in Lorentzian (indefinite signature) than Riemannian (positive definite signature) manifolds. There have been some attempts to classify them, but it seems fair to describe these attempts as preliminary and phenomenological. Commonly encountered types of curvature singularities include the following: 1. Scalar vs. nonscalar singularities: polynomial invariants like the Kretschmann scalar R_(abcd) R^(abcd) (this is the bestknown example of a degree two or quadratic invariant) may or may not blow up. Schwarzschild and Kerr vacuum: yes; ppwaves: no. Indeed, ppwaves always have identically vanishing polynomial invariants, but there are examples which nonetheless possess irremovable nonscalar curvature singularities). 2. Timelike, spacelike, null: some singular locuses in some spacetimes can be described, roughly speaking, as the limit of a family of timelike, spacelike, or null hyperslices. (This idea is most often applied in the context of a Penrose diagram used to visualize the conformal structure of a given spacetime.) For example, in the FRW models, we have a spacelike singularity. This can get tricky since unexpected things can happen: most Kerr vacuum solutions have timelike curvature singularities (inside horizons), but the Schwarzschild special case has a spacelike singularity. 3. Strong vs. weak singularities: sometimes two ideal observers whose worldlines are two different timelike curves in the same spacetime can have drastically different physical experiences as they approach the same singular locus. In weak singularities, some observers can apparently survive the encounter because roughly speaking they experience tidal forces which do indeed blow up, but so briefly that the net effect on their bodies is negligible. This can be seen in phenomena where the expansion tensor of some congruence remains finite while the curvature tensor very rapidly blows up. 4. The weakest or mildest type of irremovable singularities closely resemble certain kinds of coordinate singularities. For example, "cone singularities" occur in certain cosmic string "solutions" (look at a cone and think of the apex as a zerodimensional "reduction" of the onedimensional symmetry axis in a cylindrical coordinate chart on a manifold which is locally isometric to ordinary euclidean three dimensional space off the axis), and are also associated with unphysical "massless struts" which occur in certain axisymmetric static vacuum solutions in gtr. (See also "orbifolds" in pure mathematics.) Another example are "fold singularities" in certain colliding plane wave solutions, including the wellknown PenroseKhan CPW solution. Going back to black holes: it is true that historically one place where singularities first arose in classical field theories was in the notion of a "point mass", and at least at first sight, black hole solutions in general relativity are analogous to such point masses. However, gtr is mathematically and conceptually significantly more challenging than Maxwell's theory of EM (for example), and these analogies, while valid as far as they go, are potentially misleading. One final comment: for the topic at hand, the key difference between black holes and stars is probably that stars have a welldefined surface. (Not real stars, of course, which have a corona, stellar wind, and all that, but highly idealized stars treated as a ball of perfect fluid held together by its own gravity.) This surface is not really analogous to a black hole horizon, and it turns out that one cannot obtain, in gtr, a family of stellar models with surfaces which approach say a Schwarzschild hole, with the surfaces approaching the horizon. (Keyword: Buchdahl's theorem.) "T. Essel" 


#8
Oct1206, 05:11 AM

P: n/a

On Tue, 25 Oct 2005, Michael S. <thesparr0w1@yahoo.com> wrote:
> What properties of the big bang singularity can be inferred from the > PenroseHawking theorem? What properties of the big bang singularity > can be inferred from modern physics as a whole? By properties I mean > things like charge, mass, and spin. The problem word here is "properties". Charge, mass, and angular momentum are properties of "physical objects" like stars. In general relativity, it is not hard to define them for isolated objects (most recent textbooks explain this), but in more general situations this can get rather tricky. (Advanced students can search on the keyphrase "quasilocal mass", among other relevant topics.) The external field of a black hole (outside some compact region) will be similar or identical to that of an appropriately chosen model of an uncollapsed object like a star, so the elementary "isolated object" definitions do apply to isolated black holes. That's good enough for astrophysicists and cosmologists to talk about a "twohundred million solar mass black hole" without tying themselves into knots over fine points. Or indeed, to call such a thing an "object" on a par with objects such as stars. In a field theory, a "singularity" is basically a locus (location in a manifold) where our mathematical description of a model of some physical scenario breaks down. Sometimes our description breaks down but changing to another equally valid description (in this context, this probably means changing to another "coordinate chart") removes the problem, so these are called "removable singularities". But sometimes this doesn't work: the theory predicts some kind of behavior in the field which it just cannot handle no matter what description we attempt to employ. These are "irremovable singularities". In "metric theories" like gtr, we may encounter irremovable singularities in the curvature tensor of some spacetime. The point is that such singularities are properties of spacetime itself, not of any "physical object". To get a better sense of why the distinction is so important, we should take a closer look at singularities. Unfortunately, an enlightening general discussion is hard to come by, because curvature singularities are much harder to study in Lorentzian (indefinite signature) than Riemannian (positive definite signature) manifolds. There have been some attempts to classify them, but it seems fair to describe these attempts as preliminary and phenomenological. Commonly encountered types of curvature singularities include the following: 1. Scalar vs. nonscalar singularities: polynomial invariants like the Kretschmann scalar R_(abcd) R^(abcd) (this is the bestknown example of a degree two or quadratic invariant) may or may not blow up. Schwarzschild and Kerr vacuum: yes; ppwaves: no. Indeed, ppwaves always have identically vanishing polynomial invariants, but there are examples which nonetheless possess irremovable nonscalar curvature singularities). 2. Timelike, spacelike, null: some singular locuses in some spacetimes can be described, roughly speaking, as the limit of a family of timelike, spacelike, or null hyperslices. (This idea is most often applied in the context of a Penrose diagram used to visualize the conformal structure of a given spacetime.) For example, in the FRW models, we have a spacelike singularity. This can get tricky since unexpected things can happen: most Kerr vacuum solutions have timelike curvature singularities (inside horizons), but the Schwarzschild special case has a spacelike singularity. 3. Strong vs. weak singularities: sometimes two ideal observers whose worldlines are two different timelike curves in the same spacetime can have drastically different physical experiences as they approach the same singular locus. In weak singularities, some observers can apparently survive the encounter because roughly speaking they experience tidal forces which do indeed blow up, but so briefly that the net effect on their bodies is negligible. This can be seen in phenomena where the expansion tensor of some congruence remains finite while the curvature tensor very rapidly blows up. 4. The weakest or mildest type of irremovable singularities closely resemble certain kinds of coordinate singularities. For example, "cone singularities" occur in certain cosmic string "solutions" (look at a cone and think of the apex as a zerodimensional "reduction" of the onedimensional symmetry axis in a cylindrical coordinate chart on a manifold which is locally isometric to ordinary euclidean three dimensional space off the axis), and are also associated with unphysical "massless struts" which occur in certain axisymmetric static vacuum solutions in gtr. (See also "orbifolds" in pure mathematics.) Another example are "fold singularities" in certain colliding plane wave solutions, including the wellknown PenroseKhan CPW solution. Going back to black holes: it is true that historically one place where singularities first arose in classical field theories was in the notion of a "point mass", and at least at first sight, black hole solutions in general relativity are analogous to such point masses. However, gtr is mathematically and conceptually significantly more challenging than Maxwell's theory of EM (for example), and these analogies, while valid as far as they go, are potentially misleading. One final comment: for the topic at hand, the key difference between black holes and stars is probably that stars have a welldefined surface. (Not real stars, of course, which have a corona, stellar wind, and all that, but highly idealized stars treated as a ball of perfect fluid held together by its own gravity.) This surface is not really analogous to a black hole horizon, and it turns out that one cannot obtain, in gtr, a family of stellar models with surfaces which approach say a Schwarzschild hole, with the surfaces approaching the horizon. (Keyword: Buchdahl's theorem.) "T. Essel" 


#9
Oct1206, 05:11 AM

P: n/a

On Tue, 25 Oct 2005, Michael S. <thesparr0w1@yahoo.com> wrote:
> What properties of the big bang singularity can be inferred from the > PenroseHawking theorem? What properties of the big bang singularity > can be inferred from modern physics as a whole? By properties I mean > things like charge, mass, and spin. The problem word here is "properties". Charge, mass, and angular momentum are properties of "physical objects" like stars. In general relativity, it is not hard to define them for isolated objects (most recent textbooks explain this), but in more general situations this can get rather tricky. (Advanced students can search on the keyphrase "quasilocal mass", among other relevant topics.) The external field of a black hole (outside some compact region) will be similar or identical to that of an appropriately chosen model of an uncollapsed object like a star, so the elementary "isolated object" definitions do apply to isolated black holes. That's good enough for astrophysicists and cosmologists to talk about a "twohundred million solar mass black hole" without tying themselves into knots over fine points. Or indeed, to call such a thing an "object" on a par with objects such as stars. In a field theory, a "singularity" is basically a locus (location in a manifold) where our mathematical description of a model of some physical scenario breaks down. Sometimes our description breaks down but changing to another equally valid description (in this context, this probably means changing to another "coordinate chart") removes the problem, so these are called "removable singularities". But sometimes this doesn't work: the theory predicts some kind of behavior in the field which it just cannot handle no matter what description we attempt to employ. These are "irremovable singularities". In "metric theories" like gtr, we may encounter irremovable singularities in the curvature tensor of some spacetime. The point is that such singularities are properties of spacetime itself, not of any "physical object". To get a better sense of why the distinction is so important, we should take a closer look at singularities. Unfortunately, an enlightening general discussion is hard to come by, because curvature singularities are much harder to study in Lorentzian (indefinite signature) than Riemannian (positive definite signature) manifolds. There have been some attempts to classify them, but it seems fair to describe these attempts as preliminary and phenomenological. Commonly encountered types of curvature singularities include the following: 1. Scalar vs. nonscalar singularities: polynomial invariants like the Kretschmann scalar R_(abcd) R^(abcd) (this is the bestknown example of a degree two or quadratic invariant) may or may not blow up. Schwarzschild and Kerr vacuum: yes; ppwaves: no. Indeed, ppwaves always have identically vanishing polynomial invariants, but there are examples which nonetheless possess irremovable nonscalar curvature singularities). 2. Timelike, spacelike, null: some singular locuses in some spacetimes can be described, roughly speaking, as the limit of a family of timelike, spacelike, or null hyperslices. (This idea is most often applied in the context of a Penrose diagram used to visualize the conformal structure of a given spacetime.) For example, in the FRW models, we have a spacelike singularity. This can get tricky since unexpected things can happen: most Kerr vacuum solutions have timelike curvature singularities (inside horizons), but the Schwarzschild special case has a spacelike singularity. 3. Strong vs. weak singularities: sometimes two ideal observers whose worldlines are two different timelike curves in the same spacetime can have drastically different physical experiences as they approach the same singular locus. In weak singularities, some observers can apparently survive the encounter because roughly speaking they experience tidal forces which do indeed blow up, but so briefly that the net effect on their bodies is negligible. This can be seen in phenomena where the expansion tensor of some congruence remains finite while the curvature tensor very rapidly blows up. 4. The weakest or mildest type of irremovable singularities closely resemble certain kinds of coordinate singularities. For example, "cone singularities" occur in certain cosmic string "solutions" (look at a cone and think of the apex as a zerodimensional "reduction" of the onedimensional symmetry axis in a cylindrical coordinate chart on a manifold which is locally isometric to ordinary euclidean three dimensional space off the axis), and are also associated with unphysical "massless struts" which occur in certain axisymmetric static vacuum solutions in gtr. (See also "orbifolds" in pure mathematics.) Another example are "fold singularities" in certain colliding plane wave solutions, including the wellknown PenroseKhan CPW solution. Going back to black holes: it is true that historically one place where singularities first arose in classical field theories was in the notion of a "point mass", and at least at first sight, black hole solutions in general relativity are analogous to such point masses. However, gtr is mathematically and conceptually significantly more challenging than Maxwell's theory of EM (for example), and these analogies, while valid as far as they go, are potentially misleading. One final comment: for the topic at hand, the key difference between black holes and stars is probably that stars have a welldefined surface. (Not real stars, of course, which have a corona, stellar wind, and all that, but highly idealized stars treated as a ball of perfect fluid held together by its own gravity.) This surface is not really analogous to a black hole horizon, and it turns out that one cannot obtain, in gtr, a family of stellar models with surfaces which approach say a Schwarzschild hole, with the surfaces approaching the horizon. (Keyword: Buchdahl's theorem.) "T. Essel" 


#10
Oct1206, 05:11 AM

P: n/a

On Tue, 25 Oct 2005, Michael S. <thesparr0w1@yahoo.com> wrote:
> What properties of the big bang singularity can be inferred from the > PenroseHawking theorem? What properties of the big bang singularity > can be inferred from modern physics as a whole? By properties I mean > things like charge, mass, and spin. The problem word here is "properties". Charge, mass, and angular momentum are properties of "physical objects" like stars. In general relativity, it is not hard to define them for isolated objects (most recent textbooks explain this), but in more general situations this can get rather tricky. (Advanced students can search on the keyphrase "quasilocal mass", among other relevant topics.) The external field of a black hole (outside some compact region) will be similar or identical to that of an appropriately chosen model of an uncollapsed object like a star, so the elementary "isolated object" definitions do apply to isolated black holes. That's good enough for astrophysicists and cosmologists to talk about a "twohundred million solar mass black hole" without tying themselves into knots over fine points. Or indeed, to call such a thing an "object" on a par with objects such as stars. In a field theory, a "singularity" is basically a locus (location in a manifold) where our mathematical description of a model of some physical scenario breaks down. Sometimes our description breaks down but changing to another equally valid description (in this context, this probably means changing to another "coordinate chart") removes the problem, so these are called "removable singularities". But sometimes this doesn't work: the theory predicts some kind of behavior in the field which it just cannot handle no matter what description we attempt to employ. These are "irremovable singularities". In "metric theories" like gtr, we may encounter irremovable singularities in the curvature tensor of some spacetime. The point is that such singularities are properties of spacetime itself, not of any "physical object". To get a better sense of why the distinction is so important, we should take a closer look at singularities. Unfortunately, an enlightening general discussion is hard to come by, because curvature singularities are much harder to study in Lorentzian (indefinite signature) than Riemannian (positive definite signature) manifolds. There have been some attempts to classify them, but it seems fair to describe these attempts as preliminary and phenomenological. Commonly encountered types of curvature singularities include the following: 1. Scalar vs. nonscalar singularities: polynomial invariants like the Kretschmann scalar R_(abcd) R^(abcd) (this is the bestknown example of a degree two or quadratic invariant) may or may not blow up. Schwarzschild and Kerr vacuum: yes; ppwaves: no. Indeed, ppwaves always have identically vanishing polynomial invariants, but there are examples which nonetheless possess irremovable nonscalar curvature singularities). 2. Timelike, spacelike, null: some singular locuses in some spacetimes can be described, roughly speaking, as the limit of a family of timelike, spacelike, or null hyperslices. (This idea is most often applied in the context of a Penrose diagram used to visualize the conformal structure of a given spacetime.) For example, in the FRW models, we have a spacelike singularity. This can get tricky since unexpected things can happen: most Kerr vacuum solutions have timelike curvature singularities (inside horizons), but the Schwarzschild special case has a spacelike singularity. 3. Strong vs. weak singularities: sometimes two ideal observers whose worldlines are two different timelike curves in the same spacetime can have drastically different physical experiences as they approach the same singular locus. In weak singularities, some observers can apparently survive the encounter because roughly speaking they experience tidal forces which do indeed blow up, but so briefly that the net effect on their bodies is negligible. This can be seen in phenomena where the expansion tensor of some congruence remains finite while the curvature tensor very rapidly blows up. 4. The weakest or mildest type of irremovable singularities closely resemble certain kinds of coordinate singularities. For example, "cone singularities" occur in certain cosmic string "solutions" (look at a cone and think of the apex as a zerodimensional "reduction" of the onedimensional symmetry axis in a cylindrical coordinate chart on a manifold which is locally isometric to ordinary euclidean three dimensional space off the axis), and are also associated with unphysical "massless struts" which occur in certain axisymmetric static vacuum solutions in gtr. (See also "orbifolds" in pure mathematics.) Another example are "fold singularities" in certain colliding plane wave solutions, including the wellknown PenroseKhan CPW solution. Going back to black holes: it is true that historically one place where singularities first arose in classical field theories was in the notion of a "point mass", and at least at first sight, black hole solutions in general relativity are analogous to such point masses. However, gtr is mathematically and conceptually significantly more challenging than Maxwell's theory of EM (for example), and these analogies, while valid as far as they go, are potentially misleading. One final comment: for the topic at hand, the key difference between black holes and stars is probably that stars have a welldefined surface. (Not real stars, of course, which have a corona, stellar wind, and all that, but highly idealized stars treated as a ball of perfect fluid held together by its own gravity.) This surface is not really analogous to a black hole horizon, and it turns out that one cannot obtain, in gtr, a family of stellar models with surfaces which approach say a Schwarzschild hole, with the surfaces approaching the horizon. (Keyword: Buchdahl's theorem.) "T. Essel" 


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