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 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 it is usually preferable to solve the complete equations numerically).

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 can be used as a tool used to compare classical theories of gravitation in the limit most important for everyday gravitational experiments: the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light. PPN formalism is valid for metric theories of gravitation in which all bodies satisfy the Einstein equivalence principle (EEP). So it doesn't take into account variations in the speed of light in theories, because variations in the speed of light are not part of EEP, and PPN formalism isn't directly relevant to theories with a non-symmetric metric because it assumes that the metric is symmetric.

History

The earliest parameterizations of the post-Newtonian approximation were performed by Sir Arthur Stanley Eddington (1922). However, they dealt solely with the vacuum gravitational field outside an isolated spherical body. Dr. Ken Nordtvedt (1968, 1969) expanded this to include 7 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 10 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:$g\_\{mu\; u\}$ is the 4 by 4 symmetric metric tensor and indexes $i$ and $j$ 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 $zeta=eta=0$

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.

:$gamma=gamma$::$alpha\_1=7Delta\_1+Delta\_2-4gamma-4$:$alpha\_2=Delta\_2+zeta-1$::$zeta\_1=zeta$::::$xi$ is calculated from

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: and $alpha\_1=alpha\_2=alpha\_3=zeta\_1=zeta\_2=zeta\_3=zeta\_4=xi=0$

The mathematical relationship between the metric, metric potentials and PPN parameters for this notation is:::$g\_\{0i\}=-\; extstylefrac12(4gamma+3+alpha\_1-alpha\_2+zeta\_1-2eta)V\_i-\; extstylefrac12(1+alpha\_2-zeta\_1+2xi)W\_i-\; extstylefrac12(alpha\_1-2alpha\_2)w^iU-alpha\_2w^jU\_\{ij\};$:$g\_\{ij\}=(1+2gamma\; U)delta\_\{ij\};$

where repeated indexes are summed. $w^i$ is a velocity vector. $delta\_\{ij\}=1$ if and only if $i=j$.

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::$U=int\{\; ho\_0over|mathbf\{x\}-mathbf\{x\}\text{'}d^3x\text{'}$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 $g\_\{mu\; u\},$, scalar field $phi,$, vector field $K\_mu,$, tensor field $B\_\{mu\; u\},$ and so on; (b) prior-geometrical variables such as a flat background metric $eta\_\{mu\; u\},$, 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 $g^\{(0)\}\_\{mu\; u\}=mbox\{diag\}(-c\_0,c\_1,c\_1,c\_1),$, $phi\_0,$, $K^\{(0)\}\_mu,$, $B^\{(0)\}\_\{mu\; u\},$.

*Step 3: Get new variables from $h\_\{mu\; u\}=g\_\{mu\; u\}-g^\{(0)\}\_\{mu\; u\},$, with $phi-phi\_0,$, $K\_mu-K^\{(0)\}\_mu,$ or $B\_\{mu\; u\}-B^\{(0)\}\_\{mu\; u\},$ 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 $h\_\{mu\; u\},$. 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 $G\_\{mbox\{today\; =\; alpha/c\_0\; c\_1=1,$.

*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 $g\_\{mu\; u\},$ 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#PPN 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 Nordström's theory of gravitation the metric is given by and for this metric $gamma=-1,$, which violently disagrees with observations.

In stratified theories such as Yilmaz theory of gravitation the metric is given by and for this metric $alpha\_1=-4(gamma+1),$, which also disagrees violently with observations.

Another class of theories is the quasilinear theories such as Whitehead's theory of gravitation. For these . 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 , 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, $gamma=\; extstylefrac\{1+omega\}\{2+omega\},$. The limit of means that $omega,$ 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 vector-tensor theories. For all of these the gravitational "constant" varies with time and $alpha\_2,$ is non-zero. 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", [http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1992ApJ...393L..59W&link_type=GIF&db_key=AST&high=4221021c5705958 Astrophysical Journal, Part 2 - Letters (ISSN 0004-637X), vol. 393, no. 2, July 10, 1992, p. L59-L61.]

‡ Based on $6zeta\_4=3alpha\_3+2zeta\_1-3zeta\_3$ 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).

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.
*Nordtvedt Jr, K. (1968) Equivalence principle for massive bodies II: Theory, Phys. Rev. 169, 1017-1025.
*Nordtvedt Jr, K. (1969) Equivalence principle for massive bodies including rotational energy and radiation pressure, Phys. Rev. 180, 1293-1298.
*Will, C. M. (1971) Theoretical frameworks for testing relativistic gravity II: Parameterized post-Newtonian hydrodynamics and the Nordtvedt effect, Astrophys. J. 163, 611-628.
*Will, C.M. (1976) Active mass in relativistic gravity: Theoretical interpretation of the Kreuzer experiment, Astrophys. J., 204, 224-234.
*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, http://relativity.livingreviews.org/Articles/lrr-2006-3/
*Will, C. M., and Nordtvedt Jr., K (1972) Conservation laws and preferred frames in relativistic gravity I, The Astrophysical Journal 177, 757.

ee also

*Alternatives to general relativity#PPN parameters for a range of theories
*Linearized gravity
*Tests of general relativity
*Peskin-Takeuchi parameter The same thing as PPN, but for electroweak theory instead of gravitation