# I Is it possible to reduce or modify Einstein's Field Equations so they exactly mirror Newtonian gravity behavior?

#### haushofer

I don’t agree with this. Consider the Einstein-Infeld-Hoffman equations for approximating n-body motion in GR. These are based on the first order post Newtonian approximation. In the limit of c approaching infinite, they become exactly the Newtonian differential equation. Thus, there is a well defined Newtonian limit to a nonstatic GR situation.

Maybe I misunderstand, but how can time derivatives survive if you send c to infinity? As far as I know, there is no non-static extension of the Poisson eqn. for the Newton potential.

But I'll check your link later, maybe that clarifies some things ;)

#### PAllen

Maybe I misunderstand, but how can time derivatives survive if you send c to infinity? As far as I know, there is no non-static extension of the Poisson eqn. for the Newton potential.

But I'll check your link later, maybe that clarifies some things ;)
Yes, please check. It is pretty simple, and consistent with your links - equation of motion limit instead of field equation limit. Note that these equations are used for the most precise calculations of motion in the solar system, to calculate very small GR many body corrections to Newtonian dynamics. They form the basis of the state of the art ephemerides. Somewhat remarkable that Einstein and colleagues developed them in 1938

#### PAllen

Maybe I misunderstand, but how can time derivatives survive if you send c to infinity? As far as I know, there is no non-static extension of the Poisson eqn. for the Newton potential.

But I'll check your link later, maybe that clarifies some things ;)
Oh, I see your issue. These equations have nothing to do with the Poisson equation. They are instead related to the Newtonian N body acceleration differential equation.

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#### PAllen

Maybe I misunderstand, but how can time derivatives survive if you send c to infinity?
I don't understand this. What does the invariant speed have to do with time derivatives? In the c infinite limit, under several schemes, you have Newtonian gravity. Does that mean time derivatives can't exist in Newtonian physics??

Note, the c infinite limit of the Lorentz transform is the Galilean transform. Does that prevent time derivatives from existing???

#### PeterDonis

Mentor
how can time derivatives survive if you send c to infinity?
The Einstein-Infeld-Hoffman equations aren't differential equations. They are the Newton force equation, $G m_1 m_2 / r^2$, plus first-order post-Newtonian corrections, summed over all pairs of bodies. Since the post-Newtonian correction terms are all multiplied by $1 / c^2$, they all vanish in the limit $c \to \infty$, leaving just the Newtonian force.

#### haushofer

I don't understand this. What does the invariant speed have to do with time derivatives? In the c infinite limit, under several schemes, you have Newtonian gravity. Does that mean time derivatives can't exist in Newtonian physics??

Note, the c infinite limit of the Lorentz transform is the Galilean transform. Does that prevent time derivatives from existing???
I meant coordinate time derivatives on the metric, sorry for the confusion. These are multiplied by a factor 1\c and so constitute factors which drop out after the c-->oo limit. Of course, this has nothing to do with the derivatives in the geodesic eqn. w.r.t. the affine parameter ;)

#### PAllen

The Einstein-Infeld-Hoffman equations aren't differential equations. They are the Newton force equation, $G m_1 m_2 / r^2$, plus first-order post-Newtonian corrections, summed over all pairs of bodies. Since the post-Newtonian correction terms are all multiplied by $1 / c^2$, they all vanish in the limit $c \to \infty$, leaving just the Newtonian force.
How are they not differential equations? They relate positions and first and second derivatives thereof. For n bodies, they are a system of n complicated second order differential equations. The Newtonian case is the same, only the equations are much simpler and of course are explicitly solvable for n=2.

#### PeterDonis

Mentor
How are they not differential equations?
You're right, I was only looking at the RHS of the Newtonian case.

"Is it possible to reduce or modify Einstein's Field Equations so they exactly mirror Newtonian gravity behavior?"

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