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{{Short description|Calculational tool concerning energy, linking general relativity to classical physics}}{{General relativity sidebar |equations}}{{Being merged|spacetype=|Non-relativistic gravitational fields||section=|inactive=|date=April 2023|nocat=|dir=from}}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. In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law 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.

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:{| class="wikitable"

|How much space curvature g_{ij} is produced by unit rest mass?

	Nonlinear system>nonlinearity is there in the superposition law for gravity g_{00}?

|How much gravity is produced by unit kinetic energy textstylefrac12rho_0v^2?

	gravitational energy>gravitational potential energy rho_0/U?

|How much gravity is produced by unit internal energy rho_0Pi?

|How much gravity is produced by unit pressure p?

|Difference between radial and transverse kinetic energy on gravity

|Difference between radial and transverse stress on gravity

	Inertial frame of reference>inertial frames g_{0j} is produced by unit momentum rho_0v?

|Difference between radial and transverse momentum on dragging of inertial frames

g_{munu} is the 4 by 4 symmetric metric tensor with indexes mu and nu going from 0 to 3. Below, an index of 0 will indicate the time direction and indices i and j (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. In this notation, general relativity has PPN parameters gamma=beta=beta_1=beta_2=beta_3=beta_4=Delta_1=Delta_2=1 and zeta=eta=0.

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. The meaning of these is that alpha_1, alpha_2 and alpha_3 measure the extent of preferred frame effects. zeta_1, zeta_2, zeta_3, zeta_4 and alpha_3 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: -textstylefrac12(alpha_1-2alpha_2)w^iU-alpha_2w^jU_{ij}+O(epsilon^{frac52}) where repeated indexes are summed. epsilon is on the order of potentials such as U, the square magnitude of the coordinate velocities of matter, etc. w^i is the velocity vector of the PPN coordinate system relative to the mean rest-frame of the universe. w^2=delta_{ij}w^iw^j is the square magnitude of that velocity. delta_{ij}=1 if and only if i=j, 0 otherwise.There are ten metric potentials, U, U_{ij}, Phi_W, A, Phi_1, Phi_2, Phi_3, Phi_4, V_i and W_i, 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, where rho is the density of rest mass, Pi is the internal energy per unit rest mass, p is the pressure as measured in a local freely falling frame momentarily comoving with the matter, and mathbf{v} 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:

• Step 1: Identify the variables, which may include: (a) dynamical gravitational variables such as the metric g_{munu}, scalar field phi, vector field K_mu, tensor field B_{munu} and so on; (b) prior-geometrical variables such as a flat background metric eta_{munu}, cosmic time function t, and so on; (c) matter and non-gravitational 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 g^{(0)}_{munu} = operatorname{diag}(-c_0,c_1,c_1,c_1), phi_0, K^{(0)}_mu, B^{(0)}_{munu}.
• Step 3: Get new variables from h_{munu}=g_{munu}-g^{(0)}_{munu}, with phi-phi_0, K_mu-K^{(0)}_mu or B_{munu}-B^{(0)}_{munu} 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 h_{munu}. Substitute the perfect fluid stress tensor for the matter sources.
• Step 5: Solve for h_{00} to O(2). Assuming this tends to zero far from the system, one obtains the form h_{00}=2alpha U where U is the Newtonian gravitational potential and alpha may be a complicated function including the gravitational "constant" G. The Newtonian metric has the form g_{00}=-c_0+2alpha U, g_{0j}=0, g_{ij}=delta_{ij}c_1. Work in units where the gravitational "constant" measured today far from gravitating matter is unity so set G_{mathrm{today}} = alpha/c_0 c_1=1.
• Step 6: From linearized versions of the field equations solve for h_{ij} to O(2) and h_{0j} to O(3).
• Step 7: Solve for h_{00} to O(4). 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.
• Step 9: By comparing the result for g_{munu} 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 the metric is given by mathbf{g}=fboldsymbol{eta} and for this metric gamma=-1, which drastically disagrees with observations. In stratified theories such as Yilmaz theory of gravitation the metric is given by mathbf{g}=f_1mathbf{d}t otimes mathbf{d} t +f_2boldsymbol{eta} and for this metric alpha_1=-4(gamma+1), which also disagrees drastically with observations.Another class of theories is the quasilinear theories such as Whitehead's theory of gravitation. For these xi=beta. The relative magnitudes of the harmonics of the Earth's tides depend on xi and alpha_2, and measurements show that quasilinear theories disagree with observations of Earth's tides.Another class of metric theories is the bimetric theory. For all of these alpha_2 is non-zero. From the precession of the solar spin we know that alpha_2 < 4times 10^{-7}, and that effectively rules out bimetric theories.Another class of metric theories is the scalar–tensor theories, such as Brans–Dicke theory. For all of these, gamma=textstylefrac{1+omega}{2+omega}. The limit of gamma-1