ν β h ¯ μ α R α ν h ¯ ν α R α μ + h μ ν ( R 2 Λ ) η μ ν h α β R α β = 2 κ δ T μ ν (7)

Now impose the Hilbert-Einstein-de-Donder gauge which sets (6) to zero (fμ = 0), and suppresses any vector gravitons. Wave Equation (7) reduces to

$\begin{array}{l}{\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }-2{\stackrel{¯}{h}}_{\alpha \beta }{R}^{\alpha }{{}_{\mu \nu }}^{\beta }-{\stackrel{¯}{h}}_{\mu \alpha }{R}^{\alpha }{}_{\nu }-{\stackrel{¯}{h}}_{\nu \alpha }{R}^{\alpha }{}_{\mu }-{\eta }_{\mu \nu }{h}_{\alpha \beta }{R}^{\alpha \beta }+{h}_{\mu \nu }\left(R-2\Lambda \right)\\ =-2\kappa \delta {T}_{\mu \nu }\end{array}$ (8)

In an empty (Tμν = 0), Ricci-flat (Rμν = 0) space without Λ (R = 4Λ = 0), (8) further reduces to

${\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }-2{R}^{\alpha }{{}_{\mu \nu }}^{\beta }{\stackrel{¯}{h}}_{\alpha \beta }=-2\kappa \delta {T}_{\mu \nu }$ (9)

which is the starting point for the RWZ formalism.

Weak-Field Limit, de Sitter Metric: The Schwarzschild character of the RWZ problem above will now be relaxed, with ημν again diagonal, but M = 0 and Λ ≠ 0 in (2) and (3). The wave equation of paramount importance will follow as (17).

We know that the trace of the field Equations (1) gives $\text{4}\Lambda -R=-\kappa T$, whereby they become

${R}_{\mu \nu }-\Lambda {g}_{\mu \nu }=-\kappa \left[{T}_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }T\right]$ (10)

For an empty space (Tμν = 0 and T = 0), (10) reduces to de Sitter space

${R}_{\mu \nu }=\Lambda \text{​}\text{​}\text{ }{g}_{\mu \nu }$ (11)

and the trace to R = 4Λ.

Substitution of R and Rμν from (11) into (8) using (5) shows that the contributions due to Λ ≠ 0 are of second order in hμν. Neglecting these terms (particularly if Λ is very, very small) simplifies (8) to

${\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }-2{R}^{\alpha }{{}_{\mu \nu }}^{\beta }{\stackrel{¯}{h}}_{\alpha \beta }=-2\kappa \delta {T}_{\mu \nu }$ (12)

One can arrive at (12) to first order in hμν by using gμν as a raising and lowering operator rather than the background ημν—a result which incorrectly leads some to the conclusion that Λ terms cancel in the gravitational wave equation.

Note with caution that (12) and the RWZ Equation (9) are not the same wave equation. Overtly, the cosmological terms have vanished from (12), just like (9) where Λ was assumed in the RWZ problem to be nonexistent in the first place. However, the character of the Riemann tensor Rαμνβ is significantly different in these two relations where Λ = 0 in one but not the other.

Simplifying the SdS metric by setting the central mass M* in ημν to zero, produces the de Sitter space (11) of constant curvature K = 1/R2, where we can focus on the effect of Λ. The Riemann tensor is now

${R}_{\gamma \mu \nu \delta }=+K\left({g}_{\gamma \nu }{g}_{\mu \delta }-{g}_{\gamma \delta }{g}_{\mu \nu }\right)$ (13)

and reverts to

${R}^{\alpha }{{}_{\mu \nu }}^{\beta }=+K\left({g}^{\alpha }{}_{\nu }{g}_{\mu }{}^{\beta }-{g}^{\alpha \beta }{g}_{\mu \nu }\right)$ (14)

for use in (12). This substitution (raising and lowering with ημν) into (12) next gives K and Λ term contributions

$\begin{array}{l}-2K\left[\left({\stackrel{¯}{h}}_{\mu \nu }-{\eta }_{\mu \nu }\stackrel{¯}{h}\right)+\left({\stackrel{¯}{h}}_{\alpha \mu }{h}^{\alpha }{}_{\nu }+{\stackrel{¯}{h}}_{\nu \beta }{h}^{\beta }{}_{\mu }-\stackrel{¯}{h}{h}_{\mu \nu }-{\eta }_{\mu \nu }{h}^{\alpha \beta }{\stackrel{¯}{h}}_{\alpha \beta }\right)\right]\\ +\left[2{h}_{\mu \alpha }{\stackrel{¯}{h}}^{\alpha }{}_{\nu }+{\eta }_{\mu \nu }{h}_{\alpha \beta }^{2}\right]\end{array}$ (15)

to second order in hμν. Recalling that curvature K is related to Λ by K = Λ/3, substitution of (15) back into (12) gives to first order

${\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }-\frac{2}{3}M{\stackrel{¯}{h}}_{\mu \nu }+\frac{2}{3}\Lambda {\eta }_{\mu \nu }\stackrel{¯}{h}=-2\kappa \delta {T}_{\mu \nu }$ (16)

There is no cancellation of the Λ contributions to first order. Noting from (5) that $\stackrel{¯}{h}=h\left(1-1/2\eta \right)$, then a traceless gauge $\stackrel{¯}{h}=0$ means either that h = 0 or η = 2. Since η = 4, (16) reduces to

${\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }-\frac{2}{3}\Lambda {\stackrel{¯}{h}}_{\mu \nu }=-2\kappa \delta {T}_{\mu \nu }$ (17)

in a traceless Hilbert-Einstein-de Donder gauge where ${\stackrel{¯}{h}}_{\mu \nu }{}^{;\nu }=0$ and ${\stackrel{¯}{h}}_{\mu }{}^{\mu }=0$. (17) is a wave equation involving the Laplace-Beltrami operator term ${\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }$ for the Spin-2 gravitational perturbation ${\stackrel{¯}{h}}_{\mu \nu }$ bearing a mass

${m}_{g}=\sqrt{\text{2}\Lambda /3}$ (18)

similar to the Klein-Gordon Equation $\left(\square \text{\hspace{0.17em}}-{m}^{\text{2}}\right)\phi =0$ for a Spin-0 scalar field φ in flat Minkowski space. The Locally Flat Limit section which follows demonstrates that ${\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }\to \text{\hspace{0.17em}}\square {\stackrel{¯}{h}}_{\mu \nu }$ in (17) for the limit $r\to 0$. From (17) and (18) then

$\left(\square \text{\hspace{0.17em}}-{m}_{g}{}^{\text{2}}\right){\stackrel{¯}{h}}_{\mu \nu }=-2\kappa \delta {T}_{\mu \nu }$ (19)

in the locally flat-space limit $r\ll \text{1}$.

Note that Penrose  has pointed out that due to conformal invariance arguments, the massless Klein-Gordon equation becomes $\left(\square \text{\hspace{0.17em}}-R/6\right)\phi =0$ on a curved background. This necessarily gives (18) since R = 4Λ in de Sitter space. Also in passing, by rescaling $\stackrel{¯}{h}$ as ${h}_{2}\to 1/2{\stackrel{¯}{h}}_{1}$ in (12) and (17), then (18) becomes

${m}_{g}=\sqrt{\Lambda /3}$ (20)

which is the surface gravity κC = mg of the cosmological event horizon identified by Gibbons & Hawking . It is also found in Weinberg .

Locally Flat Limit of Wave Equation (17): It is necessary to demonstrate that hidden Λ-terms arising from ${\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }$ in (17) do not cancel the mass term in (18)-(20) when $r\to 0$ and ${\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }\to {\stackrel{¯}{h}}_{\mu \nu ,\alpha }{}^{,\alpha }=\text{\hspace{0.17em}}\square {\stackrel{¯}{h}}_{\mu \nu }$, the d’Alembertian in a locally flat region of dS studied above. Λ-terms appear but cancel out as shown below.

To simplify calculations, now note that r22 in (2) is of second-order in r and is negligible as $r\to 0$. Thus the focus is on eν (with M = 0) in (3) appearing in the diagonal of ημν and its inverse ημν. Hence, η00 = −c and η00 = −c−1, while η11 = c−1 and η11 = c. Also, note that $c\left(r\right)\to \text{1}$ and $c{\left(r\right)}^{-\text{1}}\to \text{1}$ as $r\to 0$.

Introducing the Christoffel symbol ${\Gamma }_{\alpha \beta }^{\gamma }$, we can write

${\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }={g}^{\alpha \beta }{\stackrel{¯}{h}}_{\mu \nu ;\alpha ;\beta }={g}^{\alpha \beta }\left[{\stackrel{¯}{h}}_{\mu \nu ,\alpha ;\beta }-{\left({\Gamma }_{\alpha \mu }^{\epsilon }{\stackrel{¯}{h}}_{\epsilon \nu }\right)}_{;\beta }-{\left({\Gamma }_{\alpha \nu }^{\epsilon }{\stackrel{¯}{h}}_{\mu \epsilon }\right)}_{;\beta }\right]$ (21)

Define

${\stackrel{¯}{h}}_{\mu \nu ;\alpha }{}^{;\alpha }=\text{\hspace{0.17em}}\square {\stackrel{¯}{h}}_{\mu \nu }+{A}_{\mu \nu }+{B}_{\mu \nu }+{C}_{\mu \nu }$ (22)

where

$\square {\stackrel{¯}{h}}_{\mu \nu }={\stackrel{¯}{h}}_{\mu \nu ,\alpha }{}^{,\alpha }$ (23)

${A}_{\mu \nu }=-{\Gamma }_{\beta \mu }^{\epsilon }{\stackrel{¯}{h}}_{\epsilon \nu }{}^{,\beta }-{\Gamma }_{\beta \nu }^{\epsilon }{\stackrel{¯}{h}}_{\mu \epsilon }{}^{,\beta }-{\Gamma }_{\beta \alpha }^{\epsilon }{\stackrel{¯}{h}}_{\mu \nu ,\epsilon }{\eta }^{\alpha \beta }-{\Gamma }_{\alpha \mu }^{\epsilon }{\stackrel{¯}{h}}_{\epsilon \nu }{}^{,\alpha }-{\Gamma }_{\alpha \nu }^{\epsilon }{\stackrel{¯}{h}}_{\mu \epsilon }{}^{,\alpha }$ (24)

${B}_{\mu \nu }=-{\left({\Gamma }_{\alpha \mu }^{\epsilon }\right)}^{,\alpha }{\stackrel{¯}{h}}_{\epsilon \nu }-{\left({\Gamma }_{\alpha \nu }^{\epsilon }\right)}^{,\alpha }{\stackrel{¯}{h}}_{\mu \epsilon }$ (25)

$\begin{array}{c}{C}_{\mu \nu }=-{\eta }^{\alpha \beta }\left[\left({\Gamma }_{\beta \delta }^{\epsilon }{\Gamma }_{\alpha \mu }^{\delta }-{\Gamma }_{\beta \alpha }^{\delta }{\Gamma }_{\delta \mu }^{\epsilon }-{\Gamma }_{\beta \mu }^{\delta }{\Gamma }_{\alpha \delta }^{\epsilon }\right){\stackrel{¯}{h}}_{\epsilon \nu }-{\Gamma }_{\beta \epsilon }^{\delta }{\Gamma }_{\alpha \mu }^{\epsilon }{\stackrel{¯}{h}}_{\delta \nu }-{\Gamma }_{\beta \nu }^{\delta }{\Gamma }_{\alpha \mu }^{\epsilon }{\stackrel{¯}{h}}_{\epsilon \delta }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({\Gamma }_{\beta \delta }^{\epsilon }{\Gamma }_{\alpha \nu }^{\delta }-{\Gamma }_{\beta \alpha }^{\delta }{\Gamma }_{\delta \nu }^{\epsilon }-{\Gamma }_{\beta \nu }^{\delta }{\Gamma }_{\alpha \delta }^{\epsilon }\right){\stackrel{¯}{h}}_{\mu \epsilon }-{\Gamma }_{\beta \mu }^{\delta }{\Gamma }_{\alpha \nu }^{\epsilon }{\stackrel{¯}{h}}_{\delta \epsilon }-{\Gamma }_{\beta \epsilon }^{\delta }{\Gamma }_{\alpha \nu }^{\epsilon }{\stackrel{¯}{h}}_{\mu \delta }\right]\end{array}$. (26)

Bμν is the term of interest. Aμν and Cμν contain factors of second order, or terms that vanish in locally flat space ( $r\ll \text{1}$ ). Furthermore, only the first-order second derivatives in Bμν remain as $r\to 0$. These terms are

$\begin{array}{c}{B}_{\alpha \mu \nu }^{\ast }{}^{\alpha }=-\frac{1}{2}{\eta }^{\epsilon \gamma }\left[\left({\eta }_{\alpha \gamma ,\mu }{}^{,\alpha }+{\eta }_{\mu \gamma ,\alpha }{}^{,\alpha }-{\eta }_{\alpha \mu ,\gamma }{}^{,\alpha }\right){\stackrel{¯}{h}}_{\epsilon \nu }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({\eta }_{\alpha \gamma ,\nu }{}^{,\alpha }+{\eta }_{\nu \gamma ,\alpha }{}^{,\alpha }-{\eta }_{\alpha \nu ,\gamma }{}^{,\alpha }\right){\stackrel{¯}{h}}_{\mu \epsilon }\right]\end{array}$ (27)

which can be defined as

${B}_{\alpha \mu \nu }^{\ast }{}^{\alpha }={F}_{\mu \nu }+{G}_{\mu \nu }+{H}_{\mu \nu }$ (28)

where

${F}_{\mu \nu }=-\frac{1}{2}{\eta }^{\epsilon \gamma }\left[\left(\square {\eta }_{\mu \gamma }\right){\stackrel{¯}{h}}_{\epsilon \nu }+\left(\square {\eta }_{\nu \gamma }\right){\stackrel{¯}{h}}_{\mu \epsilon }\right]$ (29)