On Sunday, September 8, 2019 at 12:47:28 AM UTC-5, Alan Grayson wrote:
>
>
>
> On Saturday, September 7, 2019 at 2:05:11 PM UTC-6, Lawrence Crowell wrote:
>>
>> On Friday, September 6, 2019 at 10:31:32 PM UTC-5, Alan Grayson wrote:
>>>
>>>
>>>
>>> On Wednesday, September 4, 2019 at 2:37:07 PM UTC-6, Lawrence Crowell
>>> wrote:
>>>>
>>>> On Wednesday, September 4, 2019 at 1:48:15 PM UTC-5, Alan Grayson wrote:
>>>>>
>>>>>
>>>>>
>>>>> On Wednesday, September 4, 2019 at 4:08:58 AM UTC-6, Lawrence Crowell
>>>>> wrote:
>>>>>>
>>>>>> You also have to include the total gravitational energy or T^{ab}
>>>>>> due to local sources and Λg^{ab}.
>>>>>>
>>>>>> The ADM Hamiltonian constraint is NH = 0 where this Hamiltonian is
>>>>>> determined by the traceless transverse part of the extrinsic curvature
>>>>>> or
>>>>>> Gauss fundamental form. For a general spacetime manifold there is no way
>>>>>> to
>>>>>> define mass-energy and for most Petrov types the mass-energy is simply
>>>>>> no
>>>>>> defined. Think of a spherical space with matter throughout. There is no
>>>>>> way
>>>>>> to construct a Gaussian surface with which to integrate a total mass or
>>>>>> energy. Also if that putative surface is embedded in mass-energy then
>>>>>> that
>>>>>> surface is subject to diffeomorphisms of local curvature. Energy is then
>>>>>> not localizable, and in general things that we want invariant are so
>>>>>> independent of such diffeomorphisms.
>>>>>>
>>>>>> LC
>>>>>>
>>>>>
>>>>> The energy of the gravitational field is positive for each particle of
>>>>> average mass. But how does one calculate the negative potential energy
>>>>> for
>>>>> each average mass particle? I can calculate the potential energy of a
>>>>> test
>>>>> particle at some location IN a field, but how can I calculate the total
>>>>> negative potential energy OF the field (for a particle of average mass)?
>>>>> AG
>>>>>
>>>>
>>>> V = -GMm/r
>>>>
>>>> Read the following where by using H = 0, zero energy and just Newtoin's
>>>> laws it is easy to derive the FLRW equations for k = 0 or a flat spatial
>>>> surface.
>>>>
>>>> LC
>>>>
>>>
>>> But if the spatial surface is flat, there is no gravity. So how can this
>>> be an argument for claiming the total estimated of a universe with gravity
>>> is zero? AG
>>>
>>
>> Not so, for it is embedded in spacetime and there is an extrinsic
>> curvature. You have to research some of this, such as reading Misner,
>> Throne & Wheeler *Gravitation* Ch 21.
>>
>> LC
>>
>
> Thanks. I have that book handly and will study your reference. However, on
> the other issue I raised, I think I am on firm ground that there is no
> general definition for the potential energy *OF* a gravitational field;
> rather just the potential energy of a test particle -- in which case
> there's something awry wih your additional of gravitation potential energy
> with rest and kinetic energy. AG
>
The definition of energy as some constant of dynamics is difficult in
general relativity.
LC
>
>>
>>>
>>>>
>>>> https://physics.stackexchange.com/questions/257476/how-did-the-universe-shift-from-dark-matter-dominated-to-dark-energy-dominate/257542#257542
>>>>
>>>>
>>>>
>>>>
>>>>>
>>>>>>
>>>>>> On Tuesday, September 3, 2019 at 10:00:55 PM UTC-5, Alan Grayson
>>>>>> wrote:
>>>>>>>
>>>>>>> Just sum over the estimated total of 10^80 particles, using mc^2 by
>>>>>>> first estimating the average mass of those particles for the rest
>>>>>>> energy,
>>>>>>> adding their average potential gravitational energy and their average
>>>>>>> kinetic energy. Why not? AG
>>>>>>>
>>>>>>
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