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 to the speed of light which in this case is better called the speed of gravity 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 is coordinate time of an observer at the test mass. The character of the functions and can be deduced by dimensional and physical analysis. The overall system depends upon the Sun's mass, and the gravitational constant , in the combination is small so the two functions can be expanded in a power series 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.

History

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.

Beta-delta notation

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: (going from 1 to 3) will indicate spatial directions.

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

Alpha-zeta notation

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 measure the failure of conservation of energy, momentum and angular momentum.

In this notation, general relativity has PPN parameters

The mathematical relationship between the metric, metric potentials and PPN parameters for this notation is:

otherwise.

, one for each PPN parameter to ensure a unique solution. 10 linear equations in 10 unknowns are solved by inverting a 10 by 10 matrix. These metric potentials have forms such as:

which is simply another way of writing the Newtonian gravitational potential,

is the coordinate velocity of the matter.

Stress-energy tensor for a perfect fluid takes form

How to apply PPN

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:

• , and so on; (c) matter and non-gravitational field variables.

• .

• if needed.

• . Substitute the perfect fluid stress tensor for the matter sources.

• .

• .

• . This is the messiest step, involving all the nonlinearities in the field equations. The stress–energy tensor must also be expanded to sufficient order.

• Step 8: Convert to local quasi-Cartesian coordinates and to standard PPN gauge.

• with the equations presented in PPN with alpha-zeta parameters, read off the PPN parameter values.

Comparisons between theories of gravity

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.

Most metric theories of gravity can be lumped into categories. Scalar theories of gravitation include conformally flat theories and stratified theories with time-orthogonal space slices.

In conformally flat theories such as Nordström's theory of gravitation, which drastically disagrees with observations. In stratified theories such as Yilmaz theory of gravitation, which also disagrees drastically with observations.

Another class of theories is the quasilinear theories such as Whitehead's theory of gravitation, and measurements show that quasilinear theories disagree with observations of Earth's tides.

Another class of metric theories is the bimetric theory, and that effectively rules out bimetric theories.

Another class of metric theories is the scalar–tensor theories, such as Brans–Dicke theory would have to be very large, so these theories are looking less and less likely as experimental accuracy improves.

, so these theories are also looking unlikely.

There are some metric theories of gravity that do not fit into the above categories, but they have similar problems.

Accuracy from experimental tests

Bounds on the PPN parameters from Will (2006) and Will (2014)† ‡ from Will (1976, 2006). It is theoretically possible from Ni (1972).

See also

• Alternatives to general relativity#Parametric post-Newtonian parameters for a range of theories

• Effective one-body formalism

• Linearized gravity

• Peskin–Takeuchi parameter The same thing as PPN, but for electroweak theory instead of gravitation

• Tests of general relativity

References

• Eddington, A. S. (1922) The Mathematical Theory of Relativity, Cambridge University Press.

• Misner, C. W., Thorne, K. S. & Wheeler, J. A. (1973) Gravitation, W. H. Freeman and Co.

• Will, C. M. (1981, 1993) Theory and Experiment in Gravitational Physics, Cambridge University Press. .

• Will, C. M., (2006) The Confrontation between General Relativity and Experiment, https://web.archive.org/web/20070613073754/http://relativity.livingreviews.org/Articles/lrr-2006-3/