In physics, precisely in the study of the theory of general relativity and many alternatives to it, the post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order deviations from 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.
Some of these post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of the matter forming the gravitational field to the speed of light, which in this case is better called the speed of gravity.
The parameterized post-Newtonian formalism or PPN formalism, is a version of this formulation that explicitly details the parameters in which a general theory of gravity can differ from Newtonian gravity.
It is used as a tool to compare Newtonian and Einsteinian 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.
The earliest parameterizations of the post-Newtonian approximation were performed by Sir Arthur Stanley Eddington in 1922.
However, they dealt solely with the vacuum gravitational field outside an isolated spherical body.
Ken Nordtvedt (1968, 1969) expanded this to include seven parameters in papers published in 1968 and 1969.
Clifford Martin Will introduced a stressed, continuous matter description of celestial bodies in 1971.
The versions described here are based on Wei-Tou Ni (1972), Will and Nordtvedt (1972), Charles W. Misner et al. (1973) (see Gravitation (book)), and Will (1981, 1993) and have ten parameters.
Ten post-Newtonian parameters completely characterize the weak-field behavior of the theory.
The formalism has been a valuable tool in tests of general relativity.
In the notation of Will (1971), Ni (1972) and Misner et al. (1973) they have the following values:
In Einstein's theory, the values of these parameters are chosen (1) to fit Newton's Law of gravity in the limit of velocities and mass approaching zero, (2) to ensure conservation of energy, mass, momentum, and angular momentum, and (3) to make the equations independent of the reference frame.
In the more recent notation of Will & Nordtvedt (1972) and Will (1981, 1993, 2006) a different set of ten PPN parameters is used.
measure the extent of preferred frame effects.
In this notation, general relativity has PPN parameters The mathematical relationship between the metric, metric potentials and PPN parameters for this notation is: where repeated indexes are summed.
, the square magnitude of the coordinate velocities of matter, etc.
is the velocity vector of the PPN coordinate system relative to the mean rest-frame of the universe.
10 linear equations in 10 unknowns are solved by inverting a 10 by 10 matrix.
is the pressure as measured in a local freely falling frame momentarily comoving with the matter, and
Stress-energy tensor for a perfect fluid takes form Examples of the process of applying PPN formalism to alternative theories of gravity can be found in Will (1981, 1993).
It is a nine step process: A table comparing PPN parameters for 23 theories of gravity can be found in Alternatives to general relativity#Parametric post-Newtonian parameters for a range of theories.
The relative magnitudes of the harmonics of the Earth's tides depend on
, and measurements show that quasilinear theories disagree with observations of Earth's tides.
Lunar laser ranging experiments tightly constrain the variation of the gravitational "constant" with time and
There are some metric theories of gravity that do not fit into the above categories, but they have similar problems.