In message http://groups.google.com/groups?dq=&...ing.google.com
I discuss the issue of tensors.

In GR the Einstein Tensor and the Stress Energy Tensor are related by
Guv=8piTuv. http://scienceworld.wolfram.com/phys...Equations.html
But then I asked, what if these two things were the same thing?

I ask you here too, because it's important. How do we really know that
there are two seperate things 'floating out there' that just HAPPEN to
be the same? What if they were in fact dual aspects of the same thing?

Wouldn't this make unifying Loop Quantum Gravity and String Theory a
lot easier if we could derive one from the other?

That, and eliminating all ad-hoc assumptions, including that causality
is unique in time at high energy scales, and that the macroscopic
fundamental constants remain unperterbed at extremely high energy
scales. There are a lot of ad hoc assumptions made by those who work
on LQG and ST. And if I ever get involved with those communities, I
plan to correct them.

(...Starblade Riven Darksquall...)

In article ,
Starblade Darksquall wrote:
In message
http://groups.google.com/groups?dq=&...ing.google.com
I discuss the issue of tensors.

In GR the Einstein Tensor and the Stress Energy Tensor are related by
Guv=8piTuv. http://scienceworld.wolfram.com/phys...Equations.html
But then I asked, what if these two things were the same thing?

I ask you here too, because it's important. How do we really know that
there are two seperate things 'floating out there' that just HAPPEN to
be the same? What if they were in fact dual aspects of the same thing?

When written

G = 8 pi T

it doesn't look very illuminating. But expand it out and you get ten
independent coupled equations, the left-hand side entirely in terms of the
metric tensor, and the right-hand side mostly in terms of masses,
pressures, and other mechanical effects. And they're infamously difficult
to find solutions for. Geometric notation hides a lot of dirty laundry.

The left hand side is the curvature tensor. It tells you in what way
space is curved. It needs to be there if you're going to have a
geometrical theory of gravity, so it's no accident that it's sitting
there. No G - no curvature - no gravity. And gravity has sources, like
mass, which are described by T, the stress-energy tensor. So again it's
no accident that T is on the right hand side. It's the moral equivalent
of

a = -GM/r^2

from Newtonian gravity. Think it's just coincidence that there just
HAPPENS to be something on the left that's equal to something on the
right?

That, and eliminating all ad-hoc assumptions, including that causality
is unique in time at high energy scales, and that the macroscopic
fundamental constants remain unperterbed at extremely high energy
scales. There are a lot of ad hoc assumptions made by those who work
on LQG and ST. And if I ever get involved with those communities, I
plan to correct them.

Nose to the grindstone, and all that.

--
"When the fool walks through the street, in his lack of understanding he
calls everything foolish." -- Ecclesiastes 10:3, New American Bible

Starblade Darksquall wrote:

In message http://groups.google.com/groups?dq=&...ing.google.com
I discuss the issue of tensors.

and we are all poorer for it.
[snip]

--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
"Quis custodiet ipsos custodes?" The Net!

Gregory L. Hansen wrote in message
...
In article ,
Starblade Darksquall wrote:
In message

http://groups.google.com/groups?dq=&...sci.physics.re
search&selm=4aa861fb.0309052011.484ae963%40posting .google.com
I discuss the issue of tensors.

In GR the Einstein Tensor and the Stress Energy Tensor are related by
Guv=8piTuv.
http://scienceworld.wolfram.com/phys...Equations.html
But then I asked, what if these two things were the same thing?

I ask you here too, because it's important. How do we really know that
there are two seperate things 'floating out there' that just HAPPEN to
be the same? What if they were in fact dual aspects of the same thing?

When written

G = 8 pi T

it doesn't look very illuminating. But expand it out and you get ten
independent coupled equations, the left-hand side entirely in terms of the
metric tensor, and the right-hand side mostly in terms of masses,
pressures, and other mechanical effects. And they're infamously difficult
to find solutions for. Geometric notation hides a lot of dirty laundry.


One of the biggest problems with teaching relativity to students is that the
compact nature of indical notation belies the fact that there's a hell of a
lot of complexity just below the surface. The 'G' in the above equation is
actually shorthand for the Ricci tensor minus one half the Ricci scalar. The
problem with this is that it's *not* given in terms of the metric tensor,
rather that it's given in terms of first and second derivatives of the
metric. Recall that the curvature tensor can be defined in terms of the
connection; this results in an expression which is quadratic in first
derivatives of the metric and linear in second derivatives. That's where the
real complexity comes from.

It's really quite funny seeing peoples' faces when they move from standard
theoretical general relativity into numerical simulations of relativity; the
boundless complexity of the equations is brought home in a crushingly
efficient manner when you begin to look at something like CactusCode thorns.

The left hand side is the curvature tensor.

Not really. In essence it's a contraction (or two ;-)) of the curvature
tensor, although I do see your point.

It tells you in what way
space is curved. It needs to be there if you're going to have a
geometrical theory of gravity, so it's no accident that it's sitting
there. No G - no curvature - no gravity.

I'd attack this from a different perspective and say that vacuum gives rise
to a vanishing Ricci scalar, but that's purely pedagogical in nature.

davidoff

"davidoff404" wrote in message ...
Gregory L. Hansen wrote in message
...
In article ,
Starblade Darksquall wrote:
In message

http://groups.google.com/groups?dq=&...sci.physics.re
search&selm=4aa861fb.0309052011.484ae963%40posting .google.com
I discuss the issue of tensors.

In GR the Einstein Tensor and the Stress Energy Tensor are related by
Guv=8piTuv.
http://scienceworld.wolfram.com/phys...Equations.html
But then I asked, what if these two things were the same thing?

I ask you here too, because it's important. How do we really know that
there are two seperate things 'floating out there' that just HAPPEN to
be the same? What if they were in fact dual aspects of the same thing?

When written

G = 8 pi T

it doesn't look very illuminating. But expand it out and you get ten
independent coupled equations, the left-hand side entirely in terms of the
metric tensor, and the right-hand side mostly in terms of masses,
pressures, and other mechanical effects. And they're infamously difficult
to find solutions for. Geometric notation hides a lot of dirty laundry.


One of the biggest problems with teaching relativity to students is that the
compact nature of indical notation belies the fact that there's a hell of a
lot of complexity just below the surface. The 'G' in the above equation is
actually shorthand for the Ricci tensor minus one half the Ricci scalar. The
problem with this is that it's *not* given in terms of the metric tensor,
rather that it's given in terms of first and second derivatives of the
metric. Recall that the curvature tensor can be defined in terms of the
connection; this results in an expression which is quadratic in first
derivatives of the metric and linear in second derivatives. That's where the
real complexity comes from.

It's really quite funny seeing peoples' faces when they move from standard
theoretical general relativity into numerical simulations of relativity; the
boundless complexity of the equations is brought home in a crushingly
efficient manner when you begin to look at something like CactusCode thorns.

The left hand side is the curvature tensor.

Not really. In essence it's a contraction (or two ;-)) of the curvature
tensor, although I do see your point.

It tells you in what way
space is curved. It needs to be there if you're going to have a
geometrical theory of gravity, so it's no accident that it's sitting
there. No G - no curvature - no gravity.

I'd attack this from a different perspective and say that vacuum gives rise
to a vanishing Ricci scalar, but that's purely pedagogical in nature.

davidoff

I think the reason we use the first and second derivitives is because
they define the relation between two different metrics. In reality, we
would not be able to tell WHICH metric we were on, but we could still
distinguish between two different metrics.

On 6 Sep 2003 18:00:37 -0700, (Starblade
Darksquall) wrote:

In message http://groups.google.com/groups?dq=&...ing.google.com
I discuss the issue of tensors.

In GR the Einstein Tensor and the Stress Energy Tensor are related by
Guv=8piTuv. http://scienceworld.wolfram.com/phys...Equations.html
But then I asked, what if these two things were the same thing?

I ask you here too, because it's important. How do we really know that
there are two seperate things 'floating out there' that just HAPPEN to
be the same? What if they were in fact dual aspects of the same thing?

Wouldn't this make unifying Loop Quantum Gravity and String Theory a
lot easier if we could derive one from the other?

That, and eliminating all ad-hoc assumptions, including that causality
is unique in time at high energy scales, and that the macroscopic
fundamental constants remain unperterbed at extremely high energy
scales. There are a lot of ad hoc assumptions made by those who work
on LQG and ST. And if I ever get involved with those communities, I
plan to correct them.

(...Starblade Riven Darksquall...)

In GR, we know precisely what form Guv has to have. Not so for Tuv.
It's quite possible that there are forms of matter and energy out
there that we haven't even yet considered and they must be squeezed
into T somehow.

Starblade Darksquall wrote in message
om...
It tells you in what way
space is curved. It needs to be there if you're going to have a
geometrical theory of gravity, so it's no accident that it's sitting
there. No G - no curvature - no gravity.

I'd attack this from a different perspective and say that vacuum gives
rise
to a vanishing Ricci scalar, but that's purely pedagogical in nature.

davidoff

I think the reason we use the first and second derivitives is because
they define the relation between two different metrics. In reality, we
would not be able to tell WHICH metric we were on, but we could still
distinguish between two different metrics.

I'm not sure quite what you mean by this. In general relativity we don't
really speak of being *on* a metric. Instead one introduces the concept of a
metric on a manifold as a means to get at important ideas like distance and
orthogonality.

Essentially, the reason one gets derivatives of the metric is because basis
vectors in the manifold vary from point to point. This is unlike the case in
flat space where the basis vectors remain constant throughout the space.
This is a pretty subtle issue at first glance, and is something I think
deserves to be illustrated with equations. As a result Usenet probably isn't
the right place to get into it. If you really want, I'll throw together a
..pdf file and give you a link to it.

davidoff

In article ,
davidoff404 wrote:

Starblade Darksquall wrote in message
. com...


Essentially, the reason one gets derivatives of the metric is because basis
vectors in the manifold vary from point to point. This is unlike the case in
flat space where the basis vectors remain constant throughout the space.

Flat space with Cartesian coordinates, anyway. Non-Cartesian coordinates
in flat space (e.g. polar coordinates) have basis vectors that change with
position. And so more work is required before you can look at a metric
and decide whether it describes something flat.
--
"When the fool walks through the street, in his lack of understanding he
calls everything foolish." -- Ecclesiastes 10:3, New American Bible

Gregory L. Hansen wrote in message
...
In article ,
davidoff404 wrote:

Starblade Darksquall wrote in message
. com...


Essentially, the reason one gets derivatives of the metric is because
basis
vectors in the manifold vary from point to point. This is unlike the case
in
flat space where the basis vectors remain constant throughout the space.

Flat space with Cartesian coordinates, anyway. Non-Cartesian coordinates
in flat space (e.g. polar coordinates) have basis vectors that change with
position. And so more work is required before you can look at a metric
and decide whether it describes something flat.


Yes, my apologies; I really have to stop being too hasty in posting replies.
I'll post a better answer tomorrow morning when I'm not so tired.

By the way, I've just been giving some thought to the original question
about why we have both first and second derivatives of the metric in the
curvature tensor and realised something interesting. I've started thinking
about using connections that are metric compatible but which are not
torsion-free and realised that some really cool stuff happens when you relax
the torsion requirement. I wonder why I never looked at that when I first
learned GR?

davidoff

"davidoff404" wrote in message ...
Gregory L. Hansen wrote in message
...
In article ,
davidoff404 wrote:

Starblade Darksquall wrote in message
. com...


Essentially, the reason one gets derivatives of the metric is because
basis
vectors in the manifold vary from point to point. This is unlike the case
in
flat space where the basis vectors remain constant throughout the space.

Flat space with Cartesian coordinates, anyway. Non-Cartesian coordinates
in flat space (e.g. polar coordinates) have basis vectors that change with
position. And so more work is required before you can look at a metric
and decide whether it describes something flat.


Yes, my apologies; I really have to stop being too hasty in posting replies.
I'll post a better answer tomorrow morning when I'm not so tired.

By the way, I've just been giving some thought to the original question
about why we have both first and second derivatives of the metric in the
curvature tensor and realised something interesting. I've started thinking
about using connections that are metric compatible but which are not
torsion-free and realised that some really cool stuff happens when you relax
the torsion requirement. I wonder why I never looked at that when I first
learned GR?

If your experience was like my continuing experience, you were trying
so desperately to follow the basic train of thought that you had
little attention to spare on those sorts of excursions. I could never
ask meaningful questions about that sort of thing on a first exposure.

Uncle Al has been going on about non-metric gravity. I'll look into
that when I get the metric stuff down.