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.
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.
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 rest-frame 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,
where is the density of rest mass, is the internal energy per unit rest mass, is the pressure as measured in a local freely falling frame momentarily comoving with the matter, and is the coordinate velocity of the matter.
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:
- 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) prior-geometrical variables such as a flat background metric , cosmic time function , 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 , , , .
- 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 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 with the equations presented in PPN with alpha-zeta parameters, read off the PPN parameter values.
Bounds on the PPN parameters from Will (2006) and Will (2014)
† Will, C. M. (10 July 1992). "Is momentum conserved? A test in the binary system PSR 1913 + 16". Astrophysical Journal Letters. 393 (2): L59–L61. Bibcode:1992ApJ...393L..59W. doi:10.1086/186451. ISSN 0004-637X.
‡ Based on from Will (1976, 2006). It is theoretically possible[clarification needed] for an alternative model of gravity to bypass this bound, in which case the bound is from Ni (1972).
- 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.
- Nordtvedt, Kenneth (1968-05-25). "Equivalence Principle for Massive Bodies. II. Theory". Physical Review. 169 (5). American Physical Society (APS): 1017–1025. Bibcode:1968PhRv..169.1017N. doi:10.1103/physrev.169.1017. ISSN 0031-899X.
- Nordtvedt, K. (1969-04-25). "Equivalence Principle for Massive Bodies Including Rotational Energy and Radiation Pressure". Physical Review. 180 (5). American Physical Society (APS): 1293–1298. Bibcode:1969PhRv..180.1293N. doi:10.1103/physrev.180.1293. ISSN 0031-899X.
- Will, Clifford M. (1971). "Theoretical Frameworks for Testing Relativistic Gravity. II. Parametrized Post-Newtonian Hydrodynamics, and the Nordtvedt Effect". The Astrophysical Journal. 163. IOP Publishing: 611-628. Bibcode:1971ApJ...163..611W. doi:10.1086/150804. ISSN 0004-637X.
- Will, C. M. (1976). "Active mass in relativistic gravity - Theoretical interpretation of the Kreuzer experiment". The Astrophysical Journal. 204. IOP Publishing: 224-234. Bibcode:1976ApJ...204..224W. doi:10.1086/154164. ISSN 0004-637X.
- Will, C. M. (1981, 1993) Theory and Experiment in Gravitational Physics, Cambridge University Press. ISBN 0-521-43973-6.
- 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/
- Will, Clifford M. (2014-06-11). "The Confrontation between General Relativity and Experiment". Living Reviews in Relativity. 17 (1): 4. arXiv:1403.7377. Bibcode:2014LRR....17....4W. doi:10.12942/lrr-2014-4. ISSN 2367-3613. PMC 5255900. PMID 28179848.
- Will, Clifford M.; Nordtvedt, Kenneth Jr. (1972). "Conservation Laws and Preferred Frames in Relativistic Gravity. I. Preferred-Frame Theories and an Extended PPN Formalism". The Astrophysical Journal. 177. IOP Publishing: 757. Bibcode:1972ApJ...177..757W. doi:10.1086/151754. ISSN 0004-637X.