Parameterized post-Newtonian formalism

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, sometimes it is 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. 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, sometimes it is 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. 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. Dr. Ken Nordtvedt (1968, 1969) expanded this to include seven parameters. Clifford Martin Will (1971) introduced a stressed, continuous matter description of celestial bodies. 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: g μ ν {displaystyle g_{mu u }} is the 4 by 4 symmetric metric tensor with indexes μ {displaystyle mu } and ν {displaystyle u } going from 0 to 3. Below, an index of 0 will indicate the time direction and indices i {displaystyle i} and j {displaystyle 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 γ = β = β 1 = β 2 = β 3 = β 4 = Δ 1 = Δ 2 = 1 {displaystyle gamma =eta =eta _{1}=eta _{2}=eta _{3}=eta _{4}=Delta _{1}=Delta _{2}=1} and ζ = η = 0 {displaystyle zeta =eta =0}