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Jonathan Scott
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I know that Newtonian gravity as a theory is linear and GR is not, but I'm trying to get a feel for why textbooks often insist relativistic gravity must be non-linear, in order to account for the gravitational effect of "gravitational energy" itself (which is however not part of the source term in GR).
What specific example could they have in mind of a case where "gravitational energy" must have an existence separate from the existing mass-energy of the objects involved?
I'm asking mainly because it seems to me that apparent non-linearity very similar to that in GR can be introduced very easily into a Newtonian-style gravitational theory as follows:
Let [itex]\phi = - Gm/r[/itex], the Newtonian potential.
Let [itex]\Phi = (1 + \phi/c^2)[/itex], the time dilation potential factor.
Suppose that what we assume to be the effective field for our relativistic equation of motion is not actually identified with the Newtonian [itex]\nabla \Phi[/itex] but rather with [itex]\Phi \, \nabla \Phi[/itex], which is of course very similar, as [itex]\Phi \approx 1[/itex].
The divergence of this effective field (giving the effective source density) is then as follows:
[tex]
\nabla (\Phi \, \nabla \Phi) = \Phi \, \nabla^2 \Phi + (\nabla \Phi)^2
[/tex]
This differs from the original linear Newtonian picture in two ways:
We can turn this round and check what effective time dilation potential [itex]\Phi_E[/itex] in terms of the Newtonian potential would cause the effective field to be given by [itex]\nabla \Phi_E = \Phi \, \nabla \Phi[/itex]. The result is as follows:
[tex]
\Phi_E = \frac{(1 + \Phi^2)}{2} = \left (1 + \frac{\phi}{c^2} + \frac{1}{2}\left ( \frac{\phi}{c^2} \right )^2 \right )
[/tex]
The additional term in the effective potential time dilation factor is exactly equal to the corresponding term in the isotropic form of the Schwarzschild solution, corresponding to the PPN [itex]\beta[/itex] non-linearity parameter being equal to 1, as in GR.
I think that as for electromagnetic energy, one can transform this integral into a boundary one and show that the integral over a large enough region is the same as the total amount of source mass (as seen locally, not reduced by time dilation) within the region, so despite the different identification of the field, the total energy still seems to come out right anyway.
So this seems to illustrate that a linear theory could give an illusion of non-linearity closely matching GR.
Does anyone know of a simple way to prove that gravity really must be a non-linear theory, without assuming that it's non-linear because GR says it is?
What specific example could they have in mind of a case where "gravitational energy" must have an existence separate from the existing mass-energy of the objects involved?
I'm asking mainly because it seems to me that apparent non-linearity very similar to that in GR can be introduced very easily into a Newtonian-style gravitational theory as follows:
Let [itex]\phi = - Gm/r[/itex], the Newtonian potential.
Let [itex]\Phi = (1 + \phi/c^2)[/itex], the time dilation potential factor.
Suppose that what we assume to be the effective field for our relativistic equation of motion is not actually identified with the Newtonian [itex]\nabla \Phi[/itex] but rather with [itex]\Phi \, \nabla \Phi[/itex], which is of course very similar, as [itex]\Phi \approx 1[/itex].
The divergence of this effective field (giving the effective source density) is then as follows:
[tex]
\nabla (\Phi \, \nabla \Phi) = \Phi \, \nabla^2 \Phi + (\nabla \Phi)^2
[/tex]
This differs from the original linear Newtonian picture in two ways:
- The Newtonian source mass density [itex]\nabla^2 \Phi[/itex] is now modified by the local time dilation factor, so the effective energy of the source is reduced by that factor. This implies a non-linear effect, as the gravitational source is causing a potential which is reducing its own effective strength as a gravitational source.
- There is an extra term proportional to the square of the field which describes a source density within the field.
This implies a non-linear effect, as the field is acting as an additional gravitational source.
We can turn this round and check what effective time dilation potential [itex]\Phi_E[/itex] in terms of the Newtonian potential would cause the effective field to be given by [itex]\nabla \Phi_E = \Phi \, \nabla \Phi[/itex]. The result is as follows:
[tex]
\Phi_E = \frac{(1 + \Phi^2)}{2} = \left (1 + \frac{\phi}{c^2} + \frac{1}{2}\left ( \frac{\phi}{c^2} \right )^2 \right )
[/tex]
The additional term in the effective potential time dilation factor is exactly equal to the corresponding term in the isotropic form of the Schwarzschild solution, corresponding to the PPN [itex]\beta[/itex] non-linearity parameter being equal to 1, as in GR.
I think that as for electromagnetic energy, one can transform this integral into a boundary one and show that the integral over a large enough region is the same as the total amount of source mass (as seen locally, not reduced by time dilation) within the region, so despite the different identification of the field, the total energy still seems to come out right anyway.
So this seems to illustrate that a linear theory could give an illusion of non-linearity closely matching GR.
Does anyone know of a simple way to prove that gravity really must be a non-linear theory, without assuming that it's non-linear because GR says it is?