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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

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

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