- #1

Jonathan Scott

Gold Member

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## Main Question or Discussion Point

I've recently noticed a very simple paradox in semi-Newtonian

gravity which seems to say that there must be some rest energy

in the field. However, this obviously conflicts with the GR

position. I'm wondering if anyone can spare a moment to clarify

the resolution of this paradox.

In basic Newtonian gravity, the change in potential energy of a

system when a pair of objects of mass m_1 and m_2 are moved from

a distance r_1 apart to distance r_2 (and brought to rest) is of

course:

U = - G m_1 m_2 (1/r_2 - 1/r_1)

However, both of these objects can be considered separately to

have decreased in potential energy by this amount, relative to

one another, which seems to imply a total change in the potential

energy of twice the external change. In basic Newtonian theory,

this is not a problem, as the potential is relative, and is of

course subject to an arbitrary additive constant anyway.

If we go a little further and consider clock rates and scalar

potentials, we see that the clock rate of each of the objects,

and hence its rest energy, is decreased by the factor of the form

(1 - Gm/rc^2) due to the potential of the other. This has the

effect of decreasing the total energy of each of the two objects

by the above amount of potential energy and means that the total

energy of the two objects has most definitely decreased by twice

the potential energy change of the system as a whole.

This means that in order for energy to be conserved, half of the

internal potential energy change must have been balanced by a

change elsewhere in the system. As far as I can see, the only

possible candidate for "elsewhere" is the gravitational field.

The relative amount of the energy that must be in the field (to

first order) is exactly the opposite of the (negative) amount

that would be present in a corresponding electrostatic

configuration, and therefore I assume that to complete this

semi-Newtonian model, a field energy density g^2/(8 pi G) in the

same mathematical form as the Maxwell energy density of the

electrostatic field would exactly match the requirements.

Does this seem valid? If so, how does this fit with the

standard interpretation that GR denies the existence of

rest energy within the field? I feel that although this

idea involves approximations and simplifications, I cannot

see any obvious reason why it should not carry over to GR.

Note that despite the similarities, this semi-Newtonian model has

some quite distinct differences from the Coulomb electrostatic

model. In the Coulomb theory the potential energy is taken to be

half of the integral of the charge density times potential, and

this is then mathematically shown to be equal to the integral of

the field energy density, on the assumption that the potential

tends to zero at a distance. In the semi-Newtonian gravitational

case the approximate potential is of the form (1 - Sum(Gm/rc^2))

summed for all local masses, which tends to 1 rather than 0, so

there is an extra term in the integral and it doesn't have a

factor of a half. It comes out as follows:

energy of masses within potential + 2 * energy of field

= original total amount of mass

or to match the description at the start of this note:

energy of masses within potential + energy of field

= original total amount of mass - energy extracted

Jonathan Scott

gravity which seems to say that there must be some rest energy

in the field. However, this obviously conflicts with the GR

position. I'm wondering if anyone can spare a moment to clarify

the resolution of this paradox.

In basic Newtonian gravity, the change in potential energy of a

system when a pair of objects of mass m_1 and m_2 are moved from

a distance r_1 apart to distance r_2 (and brought to rest) is of

course:

U = - G m_1 m_2 (1/r_2 - 1/r_1)

However, both of these objects can be considered separately to

have decreased in potential energy by this amount, relative to

one another, which seems to imply a total change in the potential

energy of twice the external change. In basic Newtonian theory,

this is not a problem, as the potential is relative, and is of

course subject to an arbitrary additive constant anyway.

If we go a little further and consider clock rates and scalar

potentials, we see that the clock rate of each of the objects,

and hence its rest energy, is decreased by the factor of the form

(1 - Gm/rc^2) due to the potential of the other. This has the

effect of decreasing the total energy of each of the two objects

by the above amount of potential energy and means that the total

energy of the two objects has most definitely decreased by twice

the potential energy change of the system as a whole.

This means that in order for energy to be conserved, half of the

internal potential energy change must have been balanced by a

change elsewhere in the system. As far as I can see, the only

possible candidate for "elsewhere" is the gravitational field.

The relative amount of the energy that must be in the field (to

first order) is exactly the opposite of the (negative) amount

that would be present in a corresponding electrostatic

configuration, and therefore I assume that to complete this

semi-Newtonian model, a field energy density g^2/(8 pi G) in the

same mathematical form as the Maxwell energy density of the

electrostatic field would exactly match the requirements.

Does this seem valid? If so, how does this fit with the

standard interpretation that GR denies the existence of

rest energy within the field? I feel that although this

idea involves approximations and simplifications, I cannot

see any obvious reason why it should not carry over to GR.

Note that despite the similarities, this semi-Newtonian model has

some quite distinct differences from the Coulomb electrostatic

model. In the Coulomb theory the potential energy is taken to be

half of the integral of the charge density times potential, and

this is then mathematically shown to be equal to the integral of

the field energy density, on the assumption that the potential

tends to zero at a distance. In the semi-Newtonian gravitational

case the approximate potential is of the form (1 - Sum(Gm/rc^2))

summed for all local masses, which tends to 1 rather than 0, so

there is an extra term in the integral and it doesn't have a

factor of a half. It comes out as follows:

energy of masses within potential + 2 * energy of field

= original total amount of mass

or to match the description at the start of this note:

energy of masses within potential + energy of field

= original total amount of mass - energy extracted

Jonathan Scott