Gamma demystified

On Mon, 12 Sep 2011, Tom Roberts wrote:

On 9/9/11 9/9/11 - 11:30 PM, Tom Roberts wrote:
Timo Nieminen wrote:
Is there some theoretical difficulty [with GR] that turns up at small scales?

Yes -- continuity is violated by quantum processes. They also violate the
locality assumptions of GR (i.e. that the energy-momentum tensor is a field on
the manifold, but quantum processes and quantum objects cannot be localized to
points of the manifold).

I should have pointed out how quantum mechanics avoids this problem.

QM uses wavefunctions that are continuous functions on spacetime, and different
states of the system correspond to different wavefunctions. The whole structure
of the theory is different, and in QM the energy-momentum tensor is not a
function on spacetime, but rather a functional of the wavefunctions. This avoids
the continuity problem, and shows the non-localizability of quantum processes
(transition probabilities between states are related to the overlap integral of
the wavefunctions, not merely a function on the manifold; the wavefunctions
always have support over a finite volume of the manifold).

Quantum Field Theory avoids the problem in a different manner, via path
integrals. All possible paths are used in each integral, and again the
energy-momentum tensor is not simply a function on the manifold, but rather a
functional of the amplitudes from the path integrals.

Thanks, that's a useful clarification.

A further question: Why does GR require continuous derivatives? For a
classical field theory in general, we don't have any such requirement - we
just need it if (at least piecewise) if we want to write a differential
version of the field equations. Lacking continuous differentiability, or
even continuity, we can't write a differential version of the field
equations, but we can still write an integral version of the field
equations. Why is GR different?

That is, we often (always?) find that the integral version of the field
equations tells us that, lacking discontinuities in constitutive relations
or boundary conditions, our fields much be continuous and continuously
differentiable. So the discontinuity must come from outside the classical
field theory, but there's no fundamental problem with dealing with it in
the theory.

Timo Nieminen wrote in
news:Pine.LNX.4.50.1109131127270.2632-100000@localhost:

On Mon, 12 Sep 2011, Tom Roberts wrote:

On 9/9/11 9/9/11 - 11:30 PM, Tom Roberts wrote:
Timo Nieminen wrote:
Is there some theoretical difficulty [with GR] that turns up at
small scales?

Yes -- continuity is violated by quantum processes. They also
violate the locality assumptions of GR (i.e. that the
energy-momentum tensor is a field on the manifold, but quantum
processes and quantum objects cannot be localized to points of the
manifold).

I should have pointed out how quantum mechanics avoids this problem.

QM uses wavefunctions that are continuous functions on spacetime, and
different states of the system correspond to different wavefunctions.
The whole structure of the theory is different, and in QM the
energy-momentum tensor is not a function on spacetime, but rather a
functional of the wavefunctions. This avoids the continuity problem,
and shows the non-localizability of quantum processes (transition
probabilities between states are related to the overlap integral of
the wavefunctions, not merely a function on the manifold; the
wavefunctions always have support over a finite volume of the
manifold).

Quantum Field Theory avoids the problem in a different manner, via
path integrals. All possible paths are used in each integral, and
again the energy-momentum tensor is not simply a function on the
manifold, but rather a functional of the amplitudes from the path
integrals.

Thanks, that's a useful clarification.

A further question: Why does GR require continuous derivatives? For a
classical field theory in general, we don't have any such requirement
- we just need it if (at least piecewise) if we want to write a
differential version of the field equations. Lacking continuous
differentiability, or even continuity, we can't write a differential
version of the field equations, but we can still write an integral
version of the field equations. Why is GR different?

GR does and does not require continuous derivatives.

The metric requires the existence of two derivatives because every
quantity in general relativity is either a function of the metric, its'
derivatives, or the stress tensor. If you want "Riemann" or the Einstein
tensor to be well defined, the metric needs second derivatives.

Now this makes sense to me, as GR is a theory of a handful of coupled
second order PDE's.

But continunity in the stress tensor? I have a bit more of a personal
difficulty with that requirement. Discontinuous sources in field
equations, electromagnetic theory among them, have never been a problem.

All that, in my understanding, is required is the existence of integrals
of the stress tensor's components. I don't even mean integrable over a
range, I just mean 'the integrals exist'. An important point when
considering stuff like point sources other distribution-based functions.

The notion of required amounts of continuity/amount of derivatives are
all fallouts of the partial differential equation theory that lies at
the base of general relativity. That stuff is only there because you
want your PDE solutions to be unique, as well as 'existing'.

I suppose a point could be made that a discontinuous stress tensor
begets a discontinuous metric, but I am uncertain of whether there is a
problem with that. I can see the argument, in terms of 'how do you
define a coordinate system for a discontinuous metric' but then you just
cover parts of the manifold ala Schwarzschild.

I think the incompatibility lies more in the fact that while GR assumes
the manifold as a result, QFT assumes the manifold as a background that
is independent of any result.

What I mean is, QFT is _always_ going to be locally and globally Lorentz
invariant. Always. Which will never, ever, be a valid solution to the GR
field equations.


That is, we often (always?) find that the integral version of the
field equations tells us that, lacking discontinuities in constitutive
relations or boundary conditions, our fields much be continuous and
continuously differentiable. So the discontinuity must come from
outside the classical field theory, but there's no fundamental problem
with dealing with it in the theory.



The invocation of integral versions vs differential versions reminds me
rather strongly of one time during a journal club meeting a paper on the
subject of the *difference* between the integral and differential
versions of Maxwell's equations came up. To this day I honestly have no
idea if the entire group of physicists at UAF was being trolled by one
of its' own, if it was brought up to argue a point, or it was genuine
confusion.

On Sep 13, 3:38*am, Timo Nieminen wrote:
On Mon, 12 Sep 2011, Tom Roberts wrote:
On 9/9/11 9/9/11 - 11:30 PM, Tom Roberts wrote:
Timo Nieminen wrote:
Is there some theoretical difficulty [with GR] that turns up at small scales?

Yes -- continuity is violated by quantum processes. They also violate the
locality assumptions of GR (i.e. that the energy-momentum tensor is a field on
the manifold, but quantum processes and quantum objects cannot be localized to
points of the manifold).

I should have pointed out how quantum mechanics avoids this problem.

QM uses wavefunctions that are continuous functions on spacetime, and different
states of the system correspond to different wavefunctions. The whole structure
of the theory is different, and in QM the energy-momentum tensor is not a
function on spacetime, but rather a functional of the wavefunctions. This avoids
the continuity problem, and shows the non-localizability of quantum processes
(transition probabilities between states are related to the overlap integral of
the wavefunctions, not merely a function on the manifold; the wavefunctions
always have support over a finite volume of the manifold).

Quantum Field Theory avoids the problem in a different manner, via path
integrals. All possible paths are used in each integral, and again the
energy-momentum tensor is not simply a function on the manifold, but rather a
functional of the amplitudes from the path integrals.

Thanks, that's a useful clarification.

A further question: Why does GR require continuous derivatives? For a
classical field theory in general, we don't have any such requirement - we
just need it if (at least piecewise) if we want to write a differential
version of the field equations. Lacking continuous differentiability, or
even continuity, we can't write a differential version of the field
equations, but we can still write an integral version of the field
equations. Why is GR different?

That is, we often (always?) find that the integral version of the field
equations tells us that, lacking discontinuities in constitutive relations
or boundary conditions, our fields much be continuous and continuously
differentiable. So the discontinuity must come from outside the classical
field theory, but there's no fundamental problem with dealing with it in
the theory.

--------------------
The problem of GR curved space
is much deeper than you imagine

it is wrong by its basic assumption that
gravity is a property of space

you can fiddle with your mathematics to the rest of your life
but still you will not give space as just space
any dynamic properties
because
gravity
IS A PROPERTY OF MASS !!
NOT OF SPACE !!

There are no coconuts at the north pole
even if you will look for them
for the rest of your life
and your grandchildren s life

ATB
Y.Porat
-----------------------------

On 9/12/11 9/12/11 - 8:38 PM, Timo Nieminen wrote:
A further question: Why does GR require continuous derivatives? [...]

I misspoke. The requirement is on the metric, not necessarily on the e-m tensor.

No time to think about this right now....


Tom Roberts

"Tom Roberts" wrote in message
...

Timo Nieminen wrote:
A further question:
Why does GR require continuous derivatives? [...]

"Tom Roberts" wrote:
I misspoke.
The requirement is on the metric,
not necessarily on the e-m tensor.
No time to think about this right now....

hanson wrote:
.... ahahahaha...AHAHAHAHAHA.. ahahaha...
Tom, you do get 5 atta-boys for that get-away!
"elegant, smooth & useless, just like SR/GR"
Thanks for the laughs... AHAHAHA... ahaha..
ahahahanson

"hanson" wrote in message
...
|
| "Tom Roberts" wrote in message
| ...
|
| Timo Nieminen wrote:
| A further question:
| Why does GR require continuous derivatives? [...]
|
| "Tom Roberts" wrote:
| I misspoke.
| The requirement is on the metric,
| not necessarily on the e-m tensor.
| No time to think about this right now....
|
| hanson wrote:
| ... ahahahaha...AHAHAHAHAHA.. ahahaha...
| Tom, you do get 5 atta-boys for that get-away!
| "elegant, smooth & useless, just like SR/GR"
| Thanks for the laughs... AHAHAHA... ahaha..
| ahahahanson
|
Elegant and smooth, eh? Is that what you call it when a cockroach
pulls it's head back from outside the woodpile? Humpty Roberts
should STUDY and LEARN, otherwise he'll always miss his spokes.

"Androcles" wrote:
"hanson" wrote in message
...
|
| "Tom Roberts" wrote in message
| ...
|
| Timo Nieminen wrote:
| A further question:
| Why does GR require continuous derivatives? [...]
|
| "Tom Roberts" wrote:
| I misspoke.
| The requirement is on the metric,
| not necessarily on the e-m tensor.
| No time to think about this right now....
|
| hanson wrote:
| ... ahahahaha...AHAHAHAHAHA.. ahahaha...
| Tom, you do get 5 atta-boys for that get-away!
| "elegant, smooth & useless, just like SR/GR"
| Thanks for the laughs... AHAHAHA... ahaha..
| ahahahanson
|
Andro wrote:
Elegant and smooth, eh?
[[[ & useless, just like SR/GR" ]]]
Is that what you call it when a cockroach pulls
it's head back from outside the woodpile?
Humpty Roberts should STUDY and LEARN,
otherwise he'll always miss his spokes.

"hanson" wrote in message
...
|
| "Androcles" wrote:
| "hanson" wrote in message
| ...
| |
| | "Tom Roberts" wrote in message
| | ...
| |
| | Timo Nieminen wrote:
| | A further question:
| | Why does GR require continuous derivatives? [...]
| |
| | "Tom Roberts" wrote:
| | I misspoke.
| | The requirement is on the metric,
| | not necessarily on the e-m tensor.
| | No time to think about this right now....
| |
| | hanson wrote:
| | ... ahahahaha...AHAHAHAHAHA.. ahahaha...
| | Tom, you do get 5 atta-boys for that get-away!
| | "elegant, smooth & useless, just like SR/GR"
| | Thanks for the laughs... AHAHAHA... ahaha..
| | ahahahanson
| |
| Andro wrote:
| Elegant and smooth, eh?
| [[[ & useless, just like SR/GR" ]]]
| Is that what you call it when a cockroach pulls
| it's head back from outside the woodpile?
| Humpty Roberts should STUDY and LEARN,
| otherwise he'll always miss his spokes.
|
|
"Useless" would be inappropriate in the context of gaining
attaboys. How about "slippery", or even "conving", then you
can award attaboys to Phuckwit Duck, aka Porky Diaper, as well?
elegant, smooth and slippery, just like SR/GR
elegant, smooth and convicting, just like SR/GR
elegant, smooth and conniving, just like SR/GR
elephant, smooch and contradictory, just like SR/GR
Difficult to choose...