Register to reply

Big Bang singularity

by thesparr0w1@yahoo.com
Tags: bang, singularity
Share this thread:
thesparr0w1@yahoo.com
#1
Oct12-06, 05:11 AM
P: n/a
What properties of the big bang singularity can be inferred from the
Penrose-Hawking 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.

Phys.Org News Partner Physics news on Phys.org
New approach to form non-equilibrium structures
Nike krypton laser achieves spot in Guinness World Records
Unleashing the power of quantum dot triplets
tessel@um.bot
#2
Oct12-06, 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
> Penrose-Hawking 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 "two-hundred 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 best-known example of a degree two or quadratic invariant)
may or may not blow up. Schwarzschild and Kerr vacuum: yes; pp-waves: no.
Indeed, pp-waves 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 zero-dimensional "reduction" of the
one-dimensional 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 well-known Penrose-Khan 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 well-defined -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"

tessel@um.bot
#3
Oct12-06, 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
> Penrose-Hawking 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 "two-hundred 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 best-known example of a degree two or quadratic invariant)
may or may not blow up. Schwarzschild and Kerr vacuum: yes; pp-waves: no.
Indeed, pp-waves 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 zero-dimensional "reduction" of the
one-dimensional 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 well-known Penrose-Khan 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 well-defined -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"


tessel@um.bot
#4
Oct12-06, 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
> Penrose-Hawking 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 "two-hundred 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 best-known example of a degree two or quadratic invariant)
may or may not blow up. Schwarzschild and Kerr vacuum: yes; pp-waves: no.
Indeed, pp-waves 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 zero-dimensional "reduction" of the
one-dimensional 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 well-known Penrose-Khan 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 well-defined -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"

tessel@um.bot
#5
Oct12-06, 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
> Penrose-Hawking 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 "two-hundred 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 best-known example of a degree two or quadratic invariant)
may or may not blow up. Schwarzschild and Kerr vacuum: yes; pp-waves: no.
Indeed, pp-waves 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 zero-dimensional "reduction" of the
one-dimensional 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 well-known Penrose-Khan 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 well-defined -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"

tessel@um.bot
#6
Oct12-06, 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
> Penrose-Hawking 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 "two-hundred 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 best-known example of a degree two or quadratic invariant)
may or may not blow up. Schwarzschild and Kerr vacuum: yes; pp-waves: no.
Indeed, pp-waves 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 zero-dimensional "reduction" of the
one-dimensional 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 well-known Penrose-Khan 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 well-defined -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"

tessel@um.bot
#7
Oct12-06, 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
> Penrose-Hawking 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 "two-hundred 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 best-known example of a degree two or quadratic invariant)
may or may not blow up. Schwarzschild and Kerr vacuum: yes; pp-waves: no.
Indeed, pp-waves 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 zero-dimensional "reduction" of the
one-dimensional 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 well-known Penrose-Khan 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 well-defined -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"

tessel@um.bot
#8
Oct12-06, 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
> Penrose-Hawking 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 "two-hundred 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 best-known example of a degree two or quadratic invariant)
may or may not blow up. Schwarzschild and Kerr vacuum: yes; pp-waves: no.
Indeed, pp-waves 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 zero-dimensional "reduction" of the
one-dimensional 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 well-known Penrose-Khan 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 well-defined -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"

tessel@um.bot
#9
Oct12-06, 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
> Penrose-Hawking 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 "two-hundred 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 best-known example of a degree two or quadratic invariant)
may or may not blow up. Schwarzschild and Kerr vacuum: yes; pp-waves: no.
Indeed, pp-waves 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 zero-dimensional "reduction" of the
one-dimensional 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 well-known Penrose-Khan 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 well-defined -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"

tessel@um.bot
#10
Oct12-06, 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
> Penrose-Hawking 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 "two-hundred 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 best-known example of a degree two or quadratic invariant)
may or may not blow up. Schwarzschild and Kerr vacuum: yes; pp-waves: no.
Indeed, pp-waves 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 zero-dimensional "reduction" of the
one-dimensional 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 well-known Penrose-Khan 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 well-defined -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"



Register to reply

Related Discussions
Big Bang Singularity Cosmology 32
Big bang and small bang black holes Cosmology 5
Big bang and singularity General Physics 13