In physics, the parameterized post-Newtonian formalism or PPN formalism, is a version of the post-Newtonian approximation to theory of general relativity which includes adjustable parameters. It is used as a tool to compare non-linear Einsteinian gravity to simpler Newtonian gravity in the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light. In general, PPN formalism can be applied to all metric theories of gravitation in which all bodies satisfy the Einstein equivalence principle (EEP). The speed of light remains constant in PPN formalism and it assumes that the metric tensor is always symmetric.
Post-Newtonian approximations
Einstein's equations of gravity are non-linear and difficult to solve even in very simple systems. A post-Newtonian expansion rewrites the equations in a series of terms proportional to a power of a small physical ratio. For example, the ratio of the velocity of the matter forming the gravitational field, v {\displaystyle v} to the speed of light, c {\displaystyle c} which in this case is better called the speed of gravity. As long as v / c ≪ 1 {\displaystyle v/c\ll 1} , the terms in a series with powers like ( v / c ) i {\displaystyle (v/c)^{i}} will have only a few important terms. The lowest order term will be Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields, it may be preferable to solve the complete equations numerically.
Parameterization The Parameterized post-Newtonian (PPN) formulation alters the basic post-Newtonian equations by replaced the numerical coefficients with adjustable parameters. Different values of the parameters create different theories of gravity suitable for different circumstances. Examples include models of the Solar System using point particle planets, systems of point masses with electric charge, and fluids with anisotropic stresses.
Static isotropic spacetime example A simple model of the Solar System uses only the gravity of the Sun and ignores its rotation. This isotropic and static spacetime would have distances known as proper time measured by the metric:
d s 2 = − A ( r ) d t 2 + B ( r ) d r 2 + r 2 d Ω 2 . {\displaystyle ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+r^{2}d\Omega ^{2}.}
Here r {\displaystyle r} is the distance from the Sun to a test mass and t {\displaystyle t} is coordinate time of an observer at the test mass. The character of the functions A and B can be deduced by dimensional and physical analysis. The overall system depends upon the Sun's mass, M and the gravitational constant G, in the combination GM. The ratio of the gravitational potential energy to rest mass of the test particle, ( G M / c 2 r ) ≪ 1 , {\displaystyle (GM/c^{2}r)\ll 1,} is small so the two functions can be expanded in a power series of that small ratio:
A ( r ) = 1 − 2 G M c 2 r + 2 ( β − γ ) ( G M c 2 r ) + … {\displaystyle A(r)=1-{\frac {2GM}{c^{2}r}}+2(\beta -\gamma ){\big (}{\frac {GM}{c^{2}r}}{\big )}+\dots }
B ( r ) = 1 + 2 γ ( G M c 2 r ) + … {\displaystyle B(r)=1+2\gamma {\big (}{\frac {GM}{c^{2}r}}{\big )}+\dots }
The parameters β {\displaystyle \beta } and γ {\displaystyle \gamma } are both equal to 1 in Einstein gravity. Solar System tests of general relativity match that value to better than one part in ten thousand. Historically other theories of gravity predicted other values but none have matched observations.
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