bar_room_physist
So I'm rolling up my sleeves and finally 'really' learning GR. I have what I assume are some misconceptions about about GR and I would love if some of you guys to chime in, call me a chuckle head, and generally point me in the right direction.

Pseudo-tensors:
***! Pseudo-tensors are not invariant! Jesus Christ doesn't that defeat the whole point?

Gravity:
Ok is it or isn't it an EM wave. The best explanation is from this quote:

QUOTE
Einstein argued that all energy has mass, and all mass acts gravitationally.  Does "gravitational energy" itself act as a source of gravity?  Now, the Einstein field equations are
Gmu,nu = 8pi Tmu,nu
Here Gmu,nu is the Einstein curvature tensor, which encodes information about the curvature of spacetime, and Tmu,nu is the so-called stress-energy tensor, which we will meet again below.  Tmu,nu represents the energy due to matter and electromagnetic fields, but includes NO contribution from "gravitational energy".  So one can argue that "gravitational energy" does NOT act as a source of gravity.  On the other hand, the Einstein field equations are non-linear; this implies that gravitational waves interact with each other (unlike light waves in Maxwell's (linear) theory).  So one can argue that "gravitational energy" IS a source of gravity.

http://math.ucr.edu/home/baez/physics/Rela.../energy_gr.html (btw it's the guy that invented the crackpot rating system, so feel free to call him a nut, you might even score him. )

curved-space time:
So if massive objects curve space time...which is really an oversimplification of the facts but close for conceptualization. Wouldn't this violate the law of least action(and perhaps the law of conservation of energy)?
If gravity works like a potential well and gravitational attraction is us 'filling' the lowest energy state. Wouldn't that imply that there is a higher field strength out side of the well..ie empty space has a field strength!!??
DaSmartest1

Holy cow

You sure bit off a big piece. General relativity is not for sissies.
waitedavid137
QUOTE (bar_room_physist+Sep 18 2011, 07:48 PM)
So I'm rolling up my sleeves and finally 'really' learning GR. I have what I assume are some misconceptions about about GR and I would love if some of you guys to chime in, call me a chuckle head, and generally point me in the right direction.

Pseudo-tensors:
***! Pseudo-tensors are not invariant! Jesus Christ doesn't that defeat the whole point?

Gravity:
Ok is it or isn't it an EM wave. The best explanation is from this quote:

http://math.ucr.edu/home/baez/physics/Rela.../energy_gr.html (btw it's the guy that invented the crackpot rating system, so feel free to call him a nut, you might even score him. )

curved-space time:
So if massive objects curve space time...which is really an oversimplification of the facts but close for conceptualization. Wouldn't this violate the law of least action(and perhaps the law of conservation of energy)?
If gravity works like a potential well and gravitational attraction is us 'filling' the lowest energy state. Wouldn't that imply that there is a higher field strength out side of the well..ie empty space has a field strength!!??

Actually its the "length" of the tensor that is invariant, its full contraction. The elements of the tensor depend on frame. This can be thought of as analogous to considering how an arrow looks when you rotate your coordinates. The mathematical description of the arrow called a vector or in this case the tensor changes its components with frame rotation, but the length stays the same, invariant to rotation. This is why Einstein and the few folk who understood relativity phrased the general principle of relativity as the laws of physics being frame covariant rather than invariant. The tensor equation representing the law of physics is indeed invariant, but the elements of the tensor which the equation ultimately relates do depend on frame, but co-vary in frame transformation in such a way that the "equation" is the same for any frame. Pseudo means something superficially looks like something that it is really not. I don't know why anyone would bother to point out how a pseudo tensor does not have the same properties as a tensor as by definition then it isn't a tensor, but I think the point is that one can define things that look like a tensor for example Christoffel symbols which actually are not tensors and that unlike a tensor, a pseudo-tensor needn't have an invariant "length". And yes if you tried to model natural laws with something other than a real tensor it would indeed defeat the point. So in GR one doesn't do that. Mass isn't the only thing that curves spacetime. Anything that you put into the stress-energy tensor will do it. If for example you have a field of massless particles such as an electromagnetic field, the length of the stress energy tensor will be zero and so will the Ricci-scalar be, but the stress-energy tensor will have nonzero components for energy pressure momentum that result in a nonzero Reimann tensor thus spacetime curvature. What is conserved in general relativity isn't Newtonian energy and momentum, but is relativistic energy and momentum parameters that correspond to isometries in the spacetime geometry.
DaSmartest1

Amazed

Very impressive. But could you tone it down to undergraduate physics level? Thank you.
bar_room_physist
Einstein realised that his field equations do not satisfy the usual conservation of energy and momentum and so, in order to save his theory from this catastrophe, he simply invented something, ad hoc, to make his theory satisfy the usual conservation laws; namely, his pseudo-tensor. Not only is this unscientific, it is also unconscionable, and completely fallacious for the following reason. Assumption of the validity of Einstein's pseudo-tensor implies the existence of a mathematical entity called a linear invariant, which is dependent solely upon the components of the metric tensor and their first derivatives. One does not even have to know the details of this; all one needs to know is that the pure mathematicians, Georgio Ricci-Curbastro and Tullio Levi-Civita, inventors of the tensor calculus, proved, in the year 1900, that such invariants do not exist! Thus, by reductio ad absurdum, Einstein's pseudo-tensor is a meaningless concoction of mathematical symbols, and consequently everything relying upon it, such as Einstein gravitational waves, is meaningless.

http://www.sjcrothers.plasmaresources.com/Levi-Civita.pdf

Paul A. M. Dirac (General Theory of Relativity, Princeton Landmarks in Physics Series, Princeton University Press, Princeton, New Jersey, 1996),
It is not possible to obtain an expression for the energy of the gravitational field satisfying both the conditions: (i) when added to other forms of energy the total energy is conserved, and (ii) the energy within a definite (three dimensional) region at a certain time is independent of the coordinate system. Thus, in general, gravitational energy cannot be localized. The best we can do is to use the pseudotensor, which satisfies condition (i) but not condition (ii). It gives us approximate information about gravitational energy, which in some special cases can be accurate.

Let us consider the energy of these waves. Owing to the pseudo-tensor not being a real tensor, we do not get, in general, a clear result independent of the coordinate system. But there is one special case in which we do get a clear result; namely, when the waves are all moving in the same direction.

from http://www.thunderbolts.info/thunderblogs/.../090807_sjc.htm

thoughts david?
waitedavid137
QUOTE (bar_room_physist+Sep 19 2011, 01:08 PM)
Einstein realized that his field equations do not satisfy the usual conservation of energy and momentum and so, in order to save his theory from this catastrophe, he simply invented something, ad hoc, to make his theory satisfy the usual conservation laws; namely, his pseudo-tensor. ...

What on earth are you talking about. He didn't invent "the pseudo-tensor" and in fact did nothing himself to show what is actually conserved in general relativity. Emmy Noether stepped in and saved the day on that one, and she didn't introduce any pseudo-tensor to do it.
At best I am left speculating on what you are referring to as "this" pseudo-tensor that you are asserting he created and my speculation on that leads me to assume you are actually talking about the pseudo-stress-energy carried by gravitational waves. At least it sounds to me like this is what you are talking about in the end of the post. Is that what you're worried about? If so, don't worry. The only exact solution known to exist as gravitational waves is for plane waves and since true plane waves won't be produced by any real things in nature we really don't know what Einstein's field equations "exactly" predicts concerning them and so can't yet compare said predictions with reality.
waitedavid137
QUOTE (DaSmartest1+Sep 19 2011, 12:35 PM)

Amazed

Very impressive. But could you tone it down to undergraduate physics level? Thank you.

I thought I did.
bar_room_physist
QUOTE (waitedavid137+Sep 20 2011, 05:08 AM)
What on earth are you talking about. He didn't invent "the pseudo-tensor" and in fact did nothing himself to show what is actually conserved in general relativity.

and i was under the impression it was Dirac that stepped in and showed energy was conserved in certain coordinate systems.

[1] ^ Albert Einstein Das hamiltonisches Prinzip und allgemeine Relativitätstheorie (The Hamiltonian principle and general relativity). Sitzungsber. preuss. Acad. Wiss. 1916, 2, 1111-1116.

[1] ^ Albert Einstein Der Energiesatz in der allgemeinen Relativitätstheorie. (An energy conservation law in general relativity). Sitzungsber. preuss. Acad. Wiss. 1918, 1, 448-459

perhaps it is better to say the "stress-energy-momentum pseudotensor" this raises serious doubts(with me at least) about advanced theoretical concepts, such as blackholes and singularities.
bar_room_physist
and as a side note david, i hope i'm not coming off as polemic or angry. I often have that effect i writing and i appreciate your input on the subject.

waitedavid137
QUOTE (bar_room_physist+Sep 19 2011, 10:17 PM)

and i was under the impression it was Dirac that stepped in and showed energy was conserved in certain coordinate systems.

[1] ^ Albert Einstein Das hamiltonisches Prinzip und allgemeine Relativitätstheorie (The Hamiltonian principle and general relativity). Sitzungsber. preuss. Acad. Wiss. 1916, 2, 1111-1116.

[1] ^ Albert Einstein Der Energiesatz in der allgemeinen Relativitätstheorie. (An energy conservation law in general relativity). Sitzungsber. preuss. Acad. Wiss. 1918, 1, 448-459

perhaps it is better to say the "stress-energy-momentum pseudotensor" this raises serious doubts(with me at least) about advanced theoretical concepts, such as blackholes and singularities.

You really should stop relying on wiki. You are looking at the pseudo stress-energy of the gravitational field concerning gravitational waves as I guessed. It shouldn't give you doubts in relativity, just in your source description. Dirac didn't save anything. It wasn't until the Noether theorem that it was understood just what was conserved in general relativity that corresponded to Newtonian energy and momentum in appropriate limits. In fact that finding of Dirac's is what causes not solves concern for how such things are conserved in general relativity.

Where it comes to gravitational radiation it obeys the exact same real tensor law as all matter does in general relativity, Einstein's field equations. What they were having trouble understanding prior to Noether is how conservation worked specifically because of the nonlinear nature of Einstein's field equations. The real stress energy tensor for gravitational radiation or waves is zero. However because of the nonlinearity, the Einstein tensor and the stress energy tensor can be thought of as a sum of expanded terms where the first order corresponds to a Newtonian potential, and stress energy acting as the source term for inhomogeneous linear wave equations now days referred to as GEM and the higher order terms of the sum would correspond to the nonlinear aspect of the Einstein tensor's expansion. The sum total of the stress-energy tensor for gravitation alone would be zero, but the first order term of its expansion would not be and is this pseudo-tensor your concerned with. So the problem for conservation was that gravitational radiation would in the Newtonian limit act as a source term and exchange energy any momentum with test particles. But how could it exchange energy and momentum with test particles if its real stress energy tensor sums to zero? You see, his pseudo-stress-energy of gravitation poses the problem for conservation. It doesn't solve it.
In history what was once defined as energy has often been revised or expanded in order to retain the concept of energy conservation. For example things in motion under friction loose energy. So the definition of energy was expanded to include heat so that energy could be conserved if one accounted for the heat. Newtonian energy any momentum is not conserved in general relativity because of exchanges with that pseudo-stress-energy of gravitational radiation. Noether was the one that found out how conservation laws work for general relativity. She found that what energy and momentum parameters are conserved in general relativity are actually what corresponds to isometries in the spacetime geometry. In the end, all things gravitational waves with a real but zero stress energy tensor, or matter with a nonzero real stress-energy tensor, whatever, obey Einstein's real tensor law of gravitation, his field equations, and what is conserved in general relativity is NOT Newtonian energy and momentum, but rather energy and momentum parameters that correspond to spacetime isometries.
bar_room_physist
QUOTE (waitedavid137+Sep 20 2011, 06:42 AM)
You really should stop relying on wiki. You are looking at the pseudo stress-energy of the gravitational field concerning gravitational waves as I guessed. It shouldn't give you doubts in relativity, just in your source description. Dirac didn't save anything. It wasn't until the Noether theorem that it was understood just what was conserved in general relativity that corresponded to Newtonian energy and momentum in appropriate limits. In fact that finding of Dirac's is what causes not solves concern for how such things are conserved in general relativity.

Where it comes to gravitational radiation it obeys the exact same real tensor law as all matter does in general relativity, Einstein's field equations. What they were having trouble understanding prior to Noether is how conservation worked specifically because of the nonlinear nature of Einstein's field equations. The real stress energy tensor for gravitational radiation or waves is zero. However because of the nonlinearity, the Einstein tensor and the stress energy tensor can be thought of as a sum of expanded terms where the first order corresponds to a Newtonian potential, and stress energy acting as the source term for inhomogeneous linear wave equations now days referred to as GEM and the higher order terms of the sum would correspond to the nonlinear aspect of the Einstein tensor's expansion. The sum total of the stress-energy tensor for gravitation alone would be zero, but the first order term of its expansion would not be and is this pseudo-tensor your concerned with. So the problem for conservation was that gravitational radiation would in the Newtonian limit act as a source term and exchange energy any momentum with test particles. But how could it exchange energy and momentum with test particles if its real stress energy tensor sums to zero? You see, his pseudo-stress-energy of gravitation poses the problem for conservation. It doesn't solve it.
In history what was once defined as energy has often been revised or expanded in order to retain the concept of energy conservation. For example things in motion under friction loose energy. So the definition of energy was expanded to include heat so that energy could be conserved if one accounted for the heat. Newtonian energy any momentum is not conserved in general relativity because of exchanges with that pseudo-stress-energy of gravitational radiation. Noether was the one that found out how conservation laws work for general relativity. She found that what energy and momentum parameters are conserved in general relativity are actually what corresponds to isometries in the spacetime geometry. In the end, all things gravitational waves with a real but zero stress energy tensor, or matter with a nonzero real stress-energy tensor, whatever, obey Einstein's real tensor law of gravitation, his field equations, and what is conserved in general relativity is NOT Newtonian energy and momentum, but rather energy and momentum parameters that correspond to spacetime isometries.

actually i was reading "Der Energiesatz in der allgemeinen Relativitätstheorie" (how i found the wikipage) but thank you for you input and your little backhanded insult.
brucep
QUOTE (bar_room_physist+Sep 20 2011, 05:17 AM)

and i was under the impression it was Dirac that stepped in and showed energy was conserved in certain coordinate systems.

[1] ^ Albert Einstein Das hamiltonisches Prinzip und allgemeine Relativitätstheorie (The Hamiltonian principle and general relativity). Sitzungsber. preuss. Acad. Wiss. 1916, 2, 1111-1116.

[1] ^ Albert Einstein Der Energiesatz in der allgemeinen Relativitätstheorie. (An energy conservation law in general relativity). Sitzungsber. preuss. Acad. Wiss. 1918, 1, 448-459

perhaps it is better to say the "stress-energy-momentum pseudotensor" this raises serious doubts(with me at least) about advanced theoretical concepts, such as blackholes and singularities.

Black holes are real natural phenomena and 'singularities' are not real natural phenomena. GR describes real natural phenomena and 'unphysical mathematical non solutions' wouldn't be considered an advanced theoretical concept in GR.
DaSmartest1

Uh oh

I smell a fight brewing. To wit, how do we know the status of black holes and singularities?
waitedavid137
QUOTE (brucep+Sep 20 2011, 04:15 AM)
Black holes are real natural phenomena and 'singularities' are not real natural phenomena. ...

How do you know this? Nonrelativistic electrodynamics describes charges as singularities as well. The only thing unique to general relativity concerning such singularities is that general relativity is able to hide them behind event horizons.
brucep
QUOTE (waitedavid137+Sep 20 2011, 12:19 PM)
How do you know this? Nonrelativistic electrodynamics describes charges as singularities as well. The only thing unique to general relativity concerning such singularities is that general relativity is able to hide them behind event horizons.

I'm convinced there's no point inside a black hole where infinite spacetime curvature is going to be a physical solution. What I base this on is the mass of the black hole can be known and it's not infinite. If they find a physical solution they'll probably still call it a singularity based on historical precedence. Let me ask you if the 'singularity' hiding behind an event horizon is coordinate independent?
waitedavid137
QUOTE (brucep+Sep 20 2011, 07:36 AM)
I'm convinced there's no point inside a black hole where infinite spacetime curvature is going to be a physical solution. What I base this on is the mass of the black hole can be known and it's not infinite. If they find a physical solution they'll probably still call it a singularity based on historical precedence. Let me ask you if the 'singularity' hiding behind an event horizon is coordinate independent?

Its not the amount of mass that is singular in classical general relativity for a black hole, it is the "density" of the mass that is singular. Who knows, maybe the correct quantum gravity theory will not have an infinite density for it. My point is that both classical electromagnetism without general relativity, and general relativity treat charges as point masses. Both treat them as having finite mass, but infinite density. Both describe the density as a singularity. The only difference is that for black holes there are event horizons around the singularity. I don't think the divergent terms in the Reimann tensor corresponding to the mass density at its location can be transformed into something finite without giving rise to other divergent terms.
brucep
QUOTE (waitedavid137+Sep 20 2011, 05:26 PM)
Its not the amount of mass that is singular in classical general relativity for a black hole, it is the "density" of the mass that is singular. Who knows, maybe the correct quantum gravity theory will not have an infinite density for it. My point is that both classical electromagnetism without general relativity, and general relativity treat charges as point masses. Both treat them as having finite mass, but infinite density. Both describe the density as a singularity. The only difference is that for black holes there are event horizons around the singularity. I don't think the divergent terms in the Riemann tensor corresponding to the mass density at its location can be transformed into something finite without giving rise to other divergent terms.

I'm aware of the point particle treatment but I never thought about treating the black hole that way. Treating a particle as a 'string' with finite extent eliminates the infinite density. String theory has it's own problem with finding a preferred vacuum from a 'huge' population of candidates. I certainly don't think GR has any problems since all this stuff is outside the GR domain of applicability. Just want to make that clear so that boundaries don't get theoretically washed over. Thank you for your comments. Have you thought about 'choosing the preferred vacuum for string theory'? The two camps of thought. One requiring empirical confirmation and the other utilizing anthropic reasoning. A bit off topic.
waitedavid137
QUOTE (brucep+Sep 20 2011, 12:02 PM)
I'm aware of the point particle treatment but I never thought about treating the black hole that way. Treating a particle as a 'string' with finite extent eliminates the infinite density. String theory has it's own problem with finding a preferred vacuum from a 'huge' population of candidates. I certainly don't think GR has any problems since all this stuff is outside the GR domain of applicability. Just want to make that clear so that boundaries don't get theoretically washed over. Thank you for your comments. Have you thought about 'choosing the preferred vacuum for string theory'? The two camps of thought. One requiring empirical confirmation and the other utilizing anthropic reasoning. A bit off topic.

I really don't put too much stock in strings yet. It seems presumptuous to me to be trying to determine which string theory, or which guage for strings really, is correct when there's no real physical evidence that strings are the correct approach to unification at all yet.
brucep
QUOTE (waitedavid137+Sep 21 2011, 11:25 AM)
I really don't put too much stock in strings yet. It seems presumptuous to me to be trying to determine which string theory, or which guage for strings really, is correct when there's no real physical evidence that strings are the correct approach to unification at all yet.

I wouldn't say presumptous since the domain of applicability is, for the most part, untestable. I take from your comment that you might think the research on quantum gravity is doomed to failure because the domain is essentially untestable?
waitedavid137
QUOTE (brucep+Sep 21 2011, 08:21 AM)
I wouldn't say presumptous since the domain of applicability is, for the most part, untestable. I take from your comment that you might think the research on quantum gravity is doomed to failure because the domain is essentially untestable?

No, I'm sure we'll come to be able to test it someday. This is still a world of savages. There is a lot of room for growth for our descendants yet.
brucep
QUOTE (waitedavid137+Sep 21 2011, 04:08 PM)
No, I'm sure we'll come to be able to test it someday. This is still a world of savages. There is a lot of room for growth for our descendants yet.

I like to hear that. Thanks for all the informative comments in this thread.
yoron
QUOTE (waitedavid137+Sep 19 2011, 05:41 AM)
Actually its the "length" of the tensor that is invariant, its full contraction. The elements of the tensor depend on frame. This can be thought of as analogous to considering how an arrow looks when you rotate your coordinates. The mathematical description of the arrow called a vector or in this case the tensor changes its components with frame rotation, but the length stays the same, invariant to rotation. This is why Einstein and the few folk who understood relativity phrased the general principle of relativity as the laws of physics being frame covariant rather than invariant. The tensor equation representing the law of physics is indeed invariant, but the elements of the tensor which the equation ultimately relates do depend on frame, but co-vary in frame transformation in such a way that the "equation" is the same for any frame. Pseudo means something superficially looks like something that it is really not. I don't know why anyone would bother to point out how a pseudo tensor does not have the same properties as a tensor as by definition then it isn't a tensor, but I think the point is that one can define things that look like a tensor for example Christoffel symbols which actually are not tensors and that unlike a tensor, a pseudo-tensor needn't have an invariant "length". And yes if you tried to model natural laws with something other than a real tensor it would indeed defeat the point. So in GR one doesn't do that. Mass isn't the only thing that curves spacetime. Anything that you put into the stress-energy tensor will do it. If for example you have a field of massless particles such as an electromagnetic field, the length of the stress energy tensor will be zero and so will the Ricci-scalar be, but the stress-energy tensor will have nonzero components for energy pressure momentum that result in a nonzero Reimann tensor thus spacetime curvature. What is conserved in general relativity isn't Newtonian energy and momentum, but is relativistic energy and momentum parameters that correspond to isometries in the spacetime geometry.