Parameterized post-Newtonian formalism

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:

${\displaystyle g_{\mu \nu }}$ is the 4 by 4 symmetric metric tensor with indexes ${\displaystyle \mu }$ and ${\displaystyle \nu }$ going from 0 to 3. Below, an index of 0 will indicate the time direction and indices ${\displaystyle i}$ and ${\displaystyle j}$ (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 ${\displaystyle \gamma =\beta ={\beta }_1={\beta }_2={\beta }_3={\beta }_4={\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=1}$ and ${\displaystyle \zeta =\eta =0.}$

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.

${\displaystyle \gamma =\gamma }$

${\displaystyle \beta =\beta }$

${\displaystyle {\alpha }_1=7{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2-4\gamma -4}$

${\displaystyle {\alpha }_2={\mathrm{\Delta }}_2+\zeta -1}$

${\displaystyle {\alpha }_3=4{\beta }_1-2\gamma -2-\zeta }$

${\displaystyle {\zeta }_1=\zeta }$

${\displaystyle {\zeta }_2=2\beta +2{\beta }_2-3\gamma -1}$

${\displaystyle {\zeta }_3={\beta }_3-1}$

${\displaystyle {\zeta }_4={\beta }_4-\gamma }$

${\displaystyle \xi }$ is calculated from ${\displaystyle 3\eta =12\beta -3\gamma -9+10\xi -3{\alpha }_1+2{\alpha }_2-2{\zeta }_1-{\zeta }_2}$

The meaning of these is that ${\displaystyle {\alpha }_1}$, ${\displaystyle {\alpha }_2}$ and ${\displaystyle {\alpha }_3}$ measure the extent of preferred frame effects. ${\displaystyle {\zeta }_1}$, ${\displaystyle {\zeta }_2}$, ${\displaystyle {\zeta }_3}$, ${\displaystyle {\zeta }_4}$ and ${\displaystyle {\alpha }_3}$ measure the failure of conservation of energy, momentum and angular momentum.

In this notation, general relativity has PPN parameters

${\displaystyle \gamma =\beta =1}$ and ${\displaystyle {\alpha }_1={\alpha }_2={\alpha }_3={\zeta }_1={\zeta }_2={\zeta }_3={\zeta }_4=\xi =0}$

The mathematical relationship between the metric, metric potentials and PPN parameters for this notation is:

${\displaystyle \begin{array}{c}{g}_{00}=-1+2U-2\beta U^2-2\xi {\mathrm{\Phi }}_W+(2\gamma +2+{\alpha }_3+{\zeta }_1-2\xi ){\mathrm{\Phi }}_1+2(3\gamma -2\beta +1+{\zeta }_2+\xi ){\mathrm{\Phi }}_2\\ +2(1+{\zeta }_3){\mathrm{\Phi }}_3+2(3\gamma +3{\zeta }_4-2\xi ){\mathrm{\Phi }}_4-({\zeta }_1-2\xi )A-({\alpha }_1-{\alpha }_2-{\alpha }_3)w^{2}U\\ -{\alpha }_2{w}^i{w}^j{U}_{ij}+(2{\alpha }_3-{\alpha }_1)w^i{V}_i+O({ϵ}^3)\end{array}}$

${\displaystyle g_{0i}=-\frac{1}{2}(4\gamma +3+{\alpha }_1-{\alpha }_2+{\zeta }_1-2\xi )V_i-\frac{1}{2}(1+{\alpha }_2-{\zeta }_1+2\xi )W_i-\frac{1}{2}({\alpha }_1-2{\alpha }_2)w^{i}U-{\alpha }_2{w}^j{U}_{ij}+O({ϵ}^{\frac{5}{2}})}$

${\displaystyle g_{ij}=(1+2\gamma U){\delta }_{ij}+O({ϵ}^2)}$

where repeated indexes are summed. ${\displaystyle ϵ}$ is on the order of potentials such as ${\displaystyle U}$, the square magnitude of the coordinate velocities of matter, etc. ${\displaystyle w^i}$ is the velocity vector of the PPN coordinate system relative to the mean rest-frame of the universe. ${\displaystyle w^2={\delta }_{ij}{w}^i{w}^j}$ is the square magnitude of that velocity. ${\displaystyle {\delta }_{ij}=1}$ if and only if ${\displaystyle i=j}$, ${\displaystyle 0}$ otherwise.

There are ten metric potentials, ${\displaystyle U}$, ${\displaystyle U_{ij}}$, ${\displaystyle {\mathrm{\Phi }}_W}$, ${\displaystyle A}$, ${\displaystyle {\mathrm{\Phi }}_1}$, ${\displaystyle {\mathrm{\Phi }}_2}$, ${\displaystyle {\mathrm{\Phi }}_3}$, ${\displaystyle {\mathrm{\Phi }}_4}$, ${\displaystyle V_i}$ and ${\displaystyle W_i}$, 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:

${\displaystyle U(\mathbf{x},t)=\int \frac{\rho ({\mathbf{x}}^{\prime },t)}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}{d}^3{x}^{\prime }}$

which is simply another way of writing the Newtonian gravitational potential,

${\displaystyle U_{ij}=\int \frac{\rho ({\mathbf{x}}^{\prime },t)(x-x^{\prime }{)}_i(x-x^{\prime }{)}_j}{|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^3}{d}^3{x}^{\prime }}$

${\displaystyle {\mathrm{\Phi }}_W=\int \frac{\rho ({\mathbf{x}}^{\prime },t)\rho ({\mathbf{x}}^{″},t)(x-x^{\prime }{)}_i}{|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^3}\left(\frac{(x^{\prime }-x^{″}{)}^i}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}-\frac{(x-x^{″}{)}^i}{|{\mathbf{x}}^{\prime }-{\mathbf{x}}^{″}|}\right)d^3{x}^{\prime }{d}^3{x}^{″}}$

${\displaystyle A=\int \frac{\rho ({\mathbf{x}}^{\prime },t){\left(\mathbf{v}({\mathbf{x}}^{\prime },t)\cdot (\mathbf{x}-{\mathbf{x}}^{\prime })\right)}^2}{|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^3}{d}^3{x}^{\prime }}$

${\displaystyle {\mathrm{\Phi }}_1=\int \frac{\rho ({\mathbf{x}}^{\prime },t)\mathbf{v}({\mathbf{x}}^{\prime },t{)}^2}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}{d}^3{x}^{\prime }}$

${\displaystyle {\mathrm{\Phi }}_2=\int \frac{\rho ({\mathbf{x}}^{\prime },t)U({\mathbf{x}}^{\prime },t)}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}{d}^3{x}^{\prime }}$

${\displaystyle {\mathrm{\Phi }}_3=\int \frac{\rho ({\mathbf{x}}^{\prime },t)\mathrm{\Pi }({\mathbf{x}}^{\prime },t)}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}{d}^3{x}^{\prime }}$

${\displaystyle {\mathrm{\Phi }}_4=\int \frac{p({\mathbf{x}}^{\prime },t)}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}{d}^3{x}^{\prime }}$

${\displaystyle V_i=\int \frac{\rho ({\mathbf{x}}^{\prime },t)v({\mathbf{x}}^{\prime },t{)}_i}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}{d}^3{x}^{\prime }}$

${\displaystyle W_i=\int \frac{\rho ({\mathbf{x}}^{\prime },t)\left(\mathbf{v}({\mathbf{x}}^{\prime },t)\cdot (\mathbf{x}-{\mathbf{x}}^{\prime })\right)(x-x^{\prime }{)}_i}{|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^3}{d}^3{x}^{\prime }}$

where ${\displaystyle \rho }$ is the density of rest mass, ${\displaystyle \mathrm{\Pi }}$ is the internal energy per unit rest mass, ${\displaystyle p}$ is the pressure as measured in a local freely falling frame momentarily comoving with the matter, and ${\displaystyle \mathbf{v}}$ is the coordinate velocity of the matter.

Stress-energy tensor for a perfect fluid takes form

${\displaystyle T^{00}=\rho (1+\mathrm{\Pi }+{\mathbf{v}}^2+2U)}$

${\displaystyle T^{0i}=\rho (1+\mathrm{\Pi }+{\mathbf{v}}^2+2U+p/\rho )v^i}$

${\displaystyle T^{ij}=\rho (1+\mathrm{\Pi }+{\mathbf{v}}^2+2U+p/\rho )v^i{v}^j+p{\delta }^{ij}(1-2\gamma U)}$

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 ${\displaystyle g_{\mu \nu }}$, scalar field ${\displaystyle \varphi }$, vector field ${\displaystyle K_{\mu }}$, tensor field ${\displaystyle B_{\mu \nu }}$ and so on; (b) prior-geometrical variables such as a flat background metric ${\displaystyle {\eta }_{\mu \nu }}$, cosmic time function ${\displaystyle t}$, 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 ${\displaystyle g_{\mu \nu }^{(0)}=\mathrm{diag}(-c_0,c_1,c_1,c_1)}$, ${\displaystyle {\varphi }_0}$, ${\displaystyle K_{\mu }^{(0)}}$, ${\displaystyle B_{\mu \nu }^{(0)}}$.
• Step 3: Get new variables from ${\displaystyle h_{\mu \nu }=g_{\mu \nu }-g_{\mu \nu }^{(0)}}$, with ${\displaystyle \varphi -{\varphi }_0}$, ${\displaystyle K_{\mu }-K_{\mu }^{(0)}}$ or ${\displaystyle B_{\mu \nu }-B_{\mu \nu }^{(0)}}$ 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 ${\displaystyle h_{\mu \nu }}$. Substitute the perfect fluid stress tensor for the matter sources.
• Step 5: Solve for ${\displaystyle h_{00}}$ to ${\displaystyle O(2)}$. Assuming this tends to zero far from the system, one obtains the form ${\displaystyle h_{00}=2\alpha U}$ where ${\displaystyle U}$ is the Newtonian gravitational potential and ${\displaystyle \alpha }$ may be a complicated function including the gravitational "constant" ${\displaystyle G}$. The Newtonian metric has the form ${\displaystyle g_{00}=-c_0+2\alpha U}$, ${\displaystyle g_{0j}=0}$, ${\displaystyle g_{ij}={\delta }_{ij}{c}_1}$. Work in units where the gravitational "constant" measured today far from gravitating matter is unity so set ${\displaystyle G_{\mathrm{t}\mathrm{o}\mathrm{d}\mathrm{a}\mathrm{y}}=\alpha /c_0{c}_1=1}$.
• Step 6: From linearized versions of the field equations solve for ${\displaystyle h_{ij}}$ to ${\displaystyle O(2)}$ and ${\displaystyle h_{0j}}$ to ${\displaystyle O(3)}$.
• Step 7: Solve for ${\displaystyle h_{00}}$ to ${\displaystyle O(4)}$. 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 ${\displaystyle g_{\mu \nu }}$ 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 ${\displaystyle \mathbf{g}=f\mathit{\eta }}$ and for this metric ${\displaystyle \gamma =-1}$, which drastically disagrees with observations. In stratified theories such as Yilmaz theory of gravitation the metric is given by ${\displaystyle \mathbf{g}=f_1\mathbf{d}t\otimes \mathbf{d}t+f_2\mathit{\eta }}$ and for this metric ${\displaystyle {\alpha }_1=-4(\gamma +1)}$, which also disagrees drastically with observations.

Another class of theories is the quasilinear theories such as Whitehead's theory of gravitation. For these ${\displaystyle \xi =\beta }$. The relative magnitudes of the harmonics of the Earth's tides depend on ${\displaystyle \xi }$ and ${\displaystyle {\alpha }_2}$, 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 ${\displaystyle {\alpha }_2}$ is non-zero. From the precession of the solar spin we know that ${\displaystyle {\alpha }_2<4\times {10}^{-7}}$, 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, ${\displaystyle \gamma =\frac{1+\omega }{2+\omega }}$. The limit of ${\displaystyle \gamma -1<2.3\times {10}^{-5}}$ means that ${\displaystyle \omega }$ 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 ${\displaystyle {\alpha }_2}$ is non-zero. Lunar laser ranging experiments tightly constrain the variation of the gravitational "constant" with time and ${\displaystyle {\alpha }_2<4\times {10}^{-7}}$, 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 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 ${\displaystyle 6{\zeta }_4=3{\alpha }_3+2{\zeta }_1-3{\zeta }_3}$ from Will (1976, 2006). It is theoretically possible for an alternative model of gravity to bypass this bound, in which case the bound is ${\displaystyle |{\zeta }_4|<0.4}$ from Ni (1972).