Stress-energy tensor transformation under coordinate scaling

[Sammywu]

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, $$T^\mu\nu$$ = 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
$$T^{\mu\nu}$$ = diag(1,0,0,0).

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

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

Every 60 light-minute equals to 1 light-hour. For a point as (1,1,1,1) in the $$\mu$$ 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

 

[Sammywu]

Gentlemen,

Sorry,

I intend for upper $$\mu$$ , but can't get it working.

 

[selfAdjoint]

Use braces {} around your mu and nu thus: T^{\mu\nu}.

 

[Sammywu]

Thanks.

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

Actually, it is probably more correct to write a tensor as
$$T^\mu _{abc}$$
in away such that
$$T^0 _{abc} = (1/6)*Mass*Time/Volume$$ for (a,b,c) = perm(1,2,3)
and
$$T^i _{abc} = (1/6)*Mass*Length/(Area*Time)$$ 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:

$$T^{\mu\nu} = T^\mu _{abc} \epsilon^{abc\nu}$$

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

Regards

 

[Sammywu]

Robphy has introduced this article in the thread about energy:

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

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.