- #1
Jonathan Scott
Gold Member
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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