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selfAdjoint has given a link to a really enlightening post by Steve Carlip
Here is the Carlip post
https://www.physicsforums.com/showthread.php?p=227804&posted=1#post227804
Carlip is tops. A major reliable authority in General Relativity and Quantum Gravity. Good idea to listen to what he says here.
He says that total energy is not defined in Gen Rel and he explains why
and he points to an open problem----a possible research area---about defining energy successfully if you "delocalize" things a bit. That comes at the end of his post.
It looks to me as if LQG is gradually solving some long-standing problems with Gen Rel and this is getting more and more interesting:
first getting rid of the bigbang singularity
then this year getting rid of black hole singularity
maybe soon getting rid of this problem of defining energy
the problem defining energy comes from the existence of "behavior at a point" in the theory----but this is analogous to what caused the bigbang singularity! In the classical theory the whole universe could come from an ideal mathematical point and so there seemed inevitably to be an instant when the density was infinite (the theory broke down). But in LQG version of big bang there is no infinite density---perhaps there was enough quantum uncertainty of location and so on to smooth it out. Could it be that a kind of "UV cutoff" did it, in some sense? Well, read Bojowald paper and see for yourself why the singularity goes away
Now I want to read Carlip post carefully and see WHY one cannot define total energy in Gen Rel. Then I will be able to guess how this obstacle is overcome when Gen Rel is quantized, say by LQG.
The key thing is very simple (partly because Carlip thinks so clearly, he's good!) there cannot be a total energy defined because it would have to include the energy of the gravitational field!
But at any given point (see the classical idea) one can always transform the coordinates so as to make the gravitational field be zero!
If the energy is invariant at every point then it can be nothing else than zero!
so there is no invariant definition of energy.
And in Gen Rel only things only have meaning if they are invariant under change of coordinates.
So you can only define energy if you cheat---you sneak in some preferred system of coordinates. Actually this is very nice to do and is done all the time in Cosmology. One reason Cosmology is such a wonderful subject THEY have a preferred time coordinate built into the FRW metric and the Friedmann equations model (Selah! as sA once declared)
But suppose we refrain from cheating and we refuse to somehow bring in enchanted preferred coordinates. Still the only problem is this zeroing out of the field at some ideal point!
what does it mean that there is a Planck length "UV cutoff"?
Must confess that I am befogged about this. Seems like energy should be definable when the theory is quantized.
Anyway, suppose we (I and whoever wants) look carefully at Carlip post.
Sounds like my wife is mashing an avocado in the kitchen, meaning Guacamole in another minute. have to get back to this
Here is the Carlip post
https://www.physicsforums.com/showthread.php?p=227804&posted=1#post227804
Carlip is tops. A major reliable authority in General Relativity and Quantum Gravity. Good idea to listen to what he says here.
He says that total energy is not defined in Gen Rel and he explains why
and he points to an open problem----a possible research area---about defining energy successfully if you "delocalize" things a bit. That comes at the end of his post.
It looks to me as if LQG is gradually solving some long-standing problems with Gen Rel and this is getting more and more interesting:
first getting rid of the bigbang singularity
then this year getting rid of black hole singularity
maybe soon getting rid of this problem of defining energy
the problem defining energy comes from the existence of "behavior at a point" in the theory----but this is analogous to what caused the bigbang singularity! In the classical theory the whole universe could come from an ideal mathematical point and so there seemed inevitably to be an instant when the density was infinite (the theory broke down). But in LQG version of big bang there is no infinite density---perhaps there was enough quantum uncertainty of location and so on to smooth it out. Could it be that a kind of "UV cutoff" did it, in some sense? Well, read Bojowald paper and see for yourself why the singularity goes away
Now I want to read Carlip post carefully and see WHY one cannot define total energy in Gen Rel. Then I will be able to guess how this obstacle is overcome when Gen Rel is quantized, say by LQG.
The key thing is very simple (partly because Carlip thinks so clearly, he's good!) there cannot be a total energy defined because it would have to include the energy of the gravitational field!
But at any given point (see the classical idea) one can always transform the coordinates so as to make the gravitational field be zero!
If the energy is invariant at every point then it can be nothing else than zero!
so there is no invariant definition of energy.
And in Gen Rel only things only have meaning if they are invariant under change of coordinates.
So you can only define energy if you cheat---you sneak in some preferred system of coordinates. Actually this is very nice to do and is done all the time in Cosmology. One reason Cosmology is such a wonderful subject
THEY have a preferred time coordinate built into the FRW metric and the Friedmann equations model (Selah! as sA once declared)But suppose we refrain from cheating and we refuse to somehow bring in enchanted preferred coordinates. Still the only problem is this zeroing out of the field at some ideal point!
what does it mean that there is a Planck length "UV cutoff"?
Must confess that I am befogged about this. Seems like energy should be definable when the theory is quantized.
Anyway, suppose we (I and whoever wants) look carefully at Carlip post.
Sounds like my wife is mashing an avocado in the kitchen, meaning Guacamole in another minute. have to get back to this
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