PostNewtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowestorder deviations from Newton's theory. 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 sometimes it is preferable to solve the complete equations numerically. Some of these postNewtonian 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. In the limit, when the fundamental speed of gravity becomes infinite, the postNewtonian expansion reduces to Newton's law of gravity.
The
parameterized postNewtonian 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 fieldThe gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
is weak and generated by objects moving slowly compared to the
speed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. 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 tensorIn general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
is always symmetric.
History
The earliest parameterizations of the postNewtonian approximation were performed by Sir
Arthur Stanley EddingtonSir Arthur Stanley Eddington, OM, FRS was a British astrophysicist of the early 20th century. He was also a philosopher of science and a popularizer of science...
in 1922. However, they dealt solely with the vacuum gravitational field outside an isolated spherical body. Dr.
Ken NordtvedtKenneth Leon Nordtvedt is a professor emeritus in the Physics Department at Montana State University and a senior researcher specializing in relativistic theories of gravity. He was born on April 16, 1939, in Chicago, Illinois. Nordtvedt graduated from the Massachusetts Institute of Technology and...
(1968, 1969) expanded this to include 7 parameters.
Clifford Martin WillClifford Martin Will is a Canadian born mathematical physicist who is well known for his contributions to the theory of general relativity....
(1971) introduced a stressed, continuous matter description of celestial bodies.
The versions described here are based on
WeiTou NiNi WeiTou is a Taiwanese physicist, who graduated from the Department of Physics of National Taiwan University , and got his PhD of Physics & Mathematics from California Institute of Technology...
(1972), Will and Nordtvedt (1972),
Charles W. MisnerCharles W. Misner is an American physicist and one of the authors of Gravitation. His specialties include general relativity and cosmology. His work has also provided early foundations for studies of quantum gravity and numerical relativity....
et al. (1973) (see
Gravitation (book)In physics, Gravitation is a very important reference book on Einstein's theory of gravity by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Often considered the "Bible" of General Relativity by researchers for its prominence. It is frequently called MTW after its authors' initials....
), and Will (1981, 1993) and have 10 parameters.
Betadelta notation
Ten
postNewtonian parameters completely characterize the weakfield behavior of the theory. The formalism has been a valuable tool in
tests of general relativityAt its introduction in 1915, the general theory of relativity did not have a solid empirical foundation. It was known that it correctly accounted for the "anomalous" precession of the perihelion of Mercury and on philosophical grounds it was considered satisfying that it was able to unify Newton's...
. In the notation of Will (1971), Ni (1972) and Misner et al. (1973) they have the following values:

How much space curvature is produced by unit rest mass ? 

How much nonlinearity is there in the superposition law for gravity ? 

How much gravity is produced by unit kinetic energy ? 

How much gravity is produced by unit gravitational potential energy ? 

How much gravity is produced by unit internal energy ? 

How much gravity is produced by unit pressure ? 

Difference between radial and transverse kinetic energy on gravity 

Difference between radial and transverse stress on gravity 

How much dragging of inertial frames is produced by unit momentum ? 

Difference between radial and transverse momentum on dragging of inertial frames 
is the 4 by 4 symmetric metric tensor and indexes
and
go from 1 to 3.
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 this notation, general relativity has PPN parameters
and
Alphazeta notation
In the more recent notation of Will & Nordtvedt (1972) and Will (1981, 1993, 2006) a different set of ten PPN parameters is used.
is calculated from
The meaning of these is that
,
and
measure the extent of preferred frame effects.
,
,
,
and
measure the failure of conservation of energy, momentum and angular momentum.
In this notation, general relativity has PPN parameters
and
The mathematical relationship between the metric, metric potentials and PPN parameters for this notation is:
where repeated indexes are summed.
is on the order of potentials such as
, the square magnitude of the coordinate velocities of matter, etc.
is the velocity vector of the PPN coordinate system relative to the mean restframe of the universe.
is the square magnitude of that velocity.
if and only if
,
otherwise.
There are ten metric potentials,
,
,
,
,
,
,
,
,
and
, 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.
A full list of metric potentials can be found in Misner et al. (1973), Will (1981, 1993, 2006) and in many other places.
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:
 Step 1: Identify the variables, which may include: (a) dynamical gravitational variables such as the metric , scalar field , vector field , tensor field and so on; (b) priorgeometrical variables such as a flat background metric , cosmic time function , and so on; (c) matter and nongravitational field variables.
 Step 2: Set the cosmological boundary conditions. Assume a homogeneous isotropic cosmology, with isotropic coordinates in the rest frame of the universe. A complete cosmological solution may or may not be needed. Call the results , , , .
 Step 3: Get new variables from , with , or if needed.
 Step 4: Substitute these forms into the field equations, keeping only such terms as are necessary to obtain a final consistent solution for . Substitute the perfect fluid stress tensor for the matter sources.
 Step 5: Solve for to . Assuming this tends to zero far from the system, one obtains the form where is the Newtonian gravitational potential and may be a complicated function including the gravitational "constant" . The Newtonian metric has the form , , . Work in units where the gravitational "constant" measured today far from gravitating matter is unity so set .
 Step 6: From linearized versions of the field equations solve for to and to .
 Step 7: Solve for to . This is the messiest step, involving all the nonlinearities in the field equations. The stressenergy tensor must also be expanded to sufficient order.
 Step 8: Convert to local quasiCartesian coordinates and to standard PPN gauge.
 Step 9: By comparing the result for with the equations presented in PPN with alphazeta 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#PPN parameters for a range of theories.
Most metric theories of gravity can be lumped into categories.
Scalar theories of gravitationScalar theories of gravitation are field theories of gravitation in which the gravitational field is described using a scalar field, which is required to satisfy some field equation....
include conformally flat theories and stratified theories with timeorthogonal space slices.
In conformally flat theories such
Nordström's theory of gravitationIn theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually two distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913 respectively...
the metric is given by
and for this metric
, which violently disagrees with observations. In stratified theories such as
Yilmaz theory of gravitationThe Yilmaz theory of gravitation is an attempt by Huseyin Yilmaz and his coworkers to formulate a classical field theory of gravitation which is similar to general relativity in weakfield conditions, but in which event horizons cannot appear.Yilmaz's work has been criticized on various grounds,...
the metric is given by
and for this metric
, which also disagrees violently with observations.
Another class of theories is the quasilinear theories such as
Whitehead's theory of gravitationIn theoretical physics, Whitehead's theory of gravitation was introduced by the distinguished mathematician and philosopher Alfred North Whitehead in 1922.Principal features of the theory:Whitehead's theory is said to feature a prior geometry...
. For these
. The relative magnitudes of the harmonics of the Earth's tides depend on
and
, and measurements show that quasilinear theories disagree with observations of Earth's tides.
Another class of metric theories is the
bimetric theoryBimetric theory refers to a class of modified theories of gravity in which two metric tensors are used instead of one. Often the second metric is introduced at high energies, with the implication that the speed of light may be energy dependent....
. For all of these
is nonzero. From the precession of the solar spin we know that
, and that effectively rules out bimetric theories.
Another class of metric theories is the
scalar tensor theoriesIn theoretical physics, a scalartensor theory is a theory that includes both a scalar field and a tensor field to represent a certain interaction...
, such as
BransDicke theoryIn theoretical physics, the Brans–Dicke theory of gravitation is a theoretical framework to explain gravitation. It is a wellknown competitor of Einstein's more popular theory of general relativity...
. For all of these,
. The limit of
means that
would have to be very large, so these theories are looking less and less likely as experimental accuracy improves.
The final main class of metric theories is the vectortensor theories. For all of these the gravitational "constant" varies with time and
is nonzero. Lunar laser ranging experiments tightly constrain the variation of the gravitational "constant" with time and
, 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 Will (2006)
† Will, C.M.,
Is momentum conserved? A test in the binary system PSR 1913 + 16,
Astrophysical Journal, Part 2  Letters (ISSN 0004637X), vol. 393, no. 2, July 10, 1992, p. L59L61.
‡ Based on
from Will (1976, 2006). It is theoretically possible for an alternative model of gravity to bypass this bound, in which case the bound is
from Ni (1972).
See also
 Alternatives to general relativity#PPN parameters for a range of theories
 Linearized gravity
Linearized gravity is an approximation scheme in general relativity in which the nonlinear contributions from the spacetime metric are ignored, simplifying the study of many problems while still producing useful approximate results.The method:...
 PeskinTakeuchi parameter
In particle physics, the Peskin–Takeuchi parameters are a set of three measurable quantities, called S, T, and U, that parameterize potential new physics contributions to electroweak radiative corrections...
The same thing as PPN, but for electroweak theory instead of gravitationGravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
 Tests of general relativity
At its introduction in 1915, the general theory of relativity did not have a solid empirical foundation. It was known that it correctly accounted for the "anomalous" precession of the perihelion of Mercury and on philosophical grounds it was considered satisfying that it was able to unify Newton's...