I can't see how stress-energy tensor meets the minumum tensor requirement

In summary, Robphy introduces an article that discusses the transformation of a symmetric stress tensor to a 3-form for integration. The article also mentions a 4-dimensional tensor, known as the "Bel-Robinson" momentum, which can be integrated to calculate energy. The author suggests that it may be possible to define a physics quantity related to a 4-dimensional domain, rather than a 3-dimensional hypersurface.
  • #1
Sammywu
273
0
Gentlemen,

I am sorry. I did a few typing errors here in order to put latex in and I even use 12 minute = 1 hour. This might confuse you.

Let me try to correct this.

Said , I use the simple dust model with 216,000 grams in a volume of 1 light-hour^3. So, [tex] T^\mu\nu [/tex] = diag(216000 , 0, 0, 0) in a coordinates using light-hour as the unit.

If now, I change to the unit of light-minute. The energy density shall be 216000 gram/ 60^3 = 1 gram/light-minute^3. So, in this coordinate
[tex] T^{\mu\nu} [/tex] = diag(1,0,0,0).

Now, if I use the standard tensor translation:
[tex] T^{\mu' \nu'} [/tex] = [tex] T^{\mu\nu} * \partial x^\mu' / \partial x^\mu * \partial x^\nu' / \partial x^\nu [/tex]

I will never get it right.
Rather, [tex] \partial x^\mu' / \partial x^\mu [/tex] = diag (60, 60, 60, 60).

Every 60 light-minute equals to 1 light-hour. For a point as (1,1,1,1) in the [tex] \mu [/tex] coordinate, its coordinates will be (60,60,60,60) for the light-minute coordinates.

I will have 216000*3600 for the energy density for the stress-energy tensor in the coordiante of light-minute then.

Did I do something wrong here?

If not, how do you reconcile this?

Thanks
 
Last edited:
Physics news on Phys.org
  • #2
Gentlemen,

Sorry,

I intend for upper [tex] \mu [/tex] , but can't get it working.
 
  • #3
Use braces {} around your mu and nu thus: T^{\mu\nu}.
 
  • #4
Thanks.

Any way, I figured out what's going on. It's a convenient way for you to write it and describe [tex] T^{00} [/tex] as mass density.

Actually, it is probably more correct to write a tensor as
[tex] T^\mu _{abc} [/tex]
in away such that
[tex] T^0 _{abc} = (1/6)*Mass*Time/Volume [/tex] for (a,b,c) = perm(1,2,3)
and
[tex] T^i _{abc} = (1/6)*Mass*Length/(Area*Time) [/tex] for i not= 0 and (a,b,c) = perm(0,1,2) or perm(0,1,3) or perm(0,2,3)

The above tensor could be used in integration.

And the stress-energy tensor in most article will be:

[tex] T^{\mu\nu} = T^\mu _{abc} \epsilon^{abc\nu} [/tex]

Any way a definition taking out the mass density part as a coefficient probablly will do too.

Regards
 
  • #5
Robphy has introduced this article in the thread about energy:

http://relativity.livingreviews.org...04-4/index.html [Broken]

In its page 11, EQ (6) has shown how the symmetric stress tensor needs to be transformed to a 3-form for integration, so as in EQ (7). .

In its page 26, a tensor of rank(0,4) as 'Bel-Robinson" momentum was shown how it can be integrated to become a quantity as energy.

In its page 25, one of the approach to a conserved quantity could be integration over a 4-dimensional domain. This of course will better be a tensor of either rank (0,4) or (1,4). I think.

Most of current approaches mentioned treat energy as a quantity pertained to a spacelike 3-dimensional hypersurface.

I wonder it's possible to look for a physics quantity pertained to a 4-dimensional domain.
 
Last edited by a moderator:

1. What is the minimum tensor requirement for the stress-energy tensor?

The minimum tensor requirement for the stress-energy tensor is that it must be a second-order tensor, meaning it has two indices, and it must satisfy the conservation law of energy and momentum.

2. What is the conservation law of energy and momentum?

The conservation law of energy and momentum states that the total amount of energy and momentum in a closed system remains constant over time. This means that energy and momentum cannot be created or destroyed, only transferred or converted into different forms.

3. How does the stress-energy tensor meet the minimum tensor requirement?

The stress-energy tensor meets the minimum tensor requirement by being a second-order tensor with two indices and satisfying the conservation law of energy and momentum. It represents the distribution of energy and momentum within a given region of space and time.

4. Why is it important for the stress-energy tensor to meet the minimum tensor requirement?

Meeting the minimum tensor requirement ensures that the stress-energy tensor is a well-defined mathematical object that accurately describes the energy and momentum within a system. This is crucial for understanding and predicting the behavior of physical systems.

5. Can the stress-energy tensor fail to meet the minimum tensor requirement?

Yes, the stress-energy tensor can fail to meet the minimum tensor requirement if it has more than two indices or does not satisfy the conservation law of energy and momentum. In such cases, it would not accurately represent the energy and momentum within a system and would not be a valid tool for studying and analyzing physical phenomena.

Similar threads

  • Special and General Relativity
Replies
4
Views
661
  • Special and General Relativity
Replies
11
Views
906
  • Special and General Relativity
Replies
1
Views
706
  • Special and General Relativity
Replies
2
Views
880
  • Special and General Relativity
Replies
9
Views
4K
Replies
1
Views
1K
  • Special and General Relativity
Replies
3
Views
1K
  • Special and General Relativity
Replies
2
Views
846
  • Special and General Relativity
Replies
21
Views
2K
  • Special and General Relativity
Replies
3
Views
2K
Back
Top