In order to determine the graviton mass of Einstein gravity (EG), we proceed as follows. A curved Kottler-Schwarzschild (KS) metric with Λ ≠ 0 will be applied to the Regge-Wheeler-Zerilli (RWZ) problem      representing gravitational radiation perturbations produced by a particle falling onto a large mass M. The RWZ result (Λ = 0) will be extended to the general EG problem with Λ ≠ 0 (EGΛ), in the fashion that Kottler extended the Schwarzschild metric to de Sitter space (SdS).
One begins with a small perturbative expansion of the Einstein field equations
about the known exact solution ημν where the metric tensor is , with hμν the dynamic perturbation of the background raising and lowering operator ημν. The most general spherically symmetric solution is well-known to be a Kottler-Schwarzschild (KS) metric
with , , and in spherically symmetric coordinates. Its contravariant inverse ημν is defined such that .
The wave equation for gravitational radiation hμν on the non-flat background containing Λ in (1) will follow as (9) below, derived now from the procedure developed in the RWZ formalism. Perturbation analysis of (1) for a stable background produces the following
Stability must be assumed in order that δTμν is small. This equation can be simplified by defining the function (introduced by Einstein himself)
and its divergence
Substituting (5) and (6) into (4) and re-grouping terms gives
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
In an empty (Tμν = 0), Ricci-flat (Rμν = 0) space without Λ (R = 4Λ = 0), (8) further reduces to
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 , whereby they become
For an empty space (Tμν = 0 and T = 0), (10) reduces to de Sitter space
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
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
and reverts to
for use in (12). This substitution (raising and lowering with ημν) into (12) next gives K and Λ term contributions
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
There is no cancellation of the Λ contributions to first order. Noting from (5) that , then a traceless gauge means either that h = 0 or η = 2. Since η = 4, (16) reduces to
in a traceless Hilbert-Einstein-de Donder gauge where and . (17) is a wave equation involving the Laplace-Beltrami operator term for the Spin-2 gravitational perturbation bearing a mass
similar to the Klein-Gordon Equation for a Spin-0 scalar field φ in flat Minkowski space. The Locally Flat Limit section which follows demonstrates that in (17) for the limit . From (17) and (18) then
in the locally flat-space limit .
Note that Penrose  has pointed out that due to conformal invariance arguments, the massless Klein-Gordon equation becomes on a curved background. This necessarily gives (18) since R = 4Λ in de Sitter space. Also in passing, by rescaling as in (12) and (17), then (18) becomes
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 in (17) do not cancel the mass term in (18)-(20) when and , 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 r2dΩ2 in (2) is of second-order in r and is negligible as . 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 and as .
Introducing the Christoffel symbol , we can write
Bμν is the term of interest. Aμν and Cμν contain factors of second order, or terms that vanish in locally flat space ( ). Furthermore, only the first-order second derivatives in Bμν remain as . These terms are
which can be defined as
In this approximation, . Also and .
We find that
whereby (all other terms do not contribute)
whereby (all other terms do not contribute)
; ; . (36)
; ; ; (38)
Summarizing, the two contributing terms to Fμν in (33) and (34) are equal and opposite thereby cancelling in (32). Thus, Fμν = 0. Similarly, the collective Gμν and Hμν terms in (36) and (38) cancel one another, giving Gμν + Hμν = 0. Hence in (28) and (25). Therefore we get in the locally flat limit of (17).
The graviton mass (18) for EGΛ thus follows from this analysis, a result first determined many years ago .
Identifying Einstein Gravity as a Partially Massless Theory: The cosmological phase diagrams for partially massless fields of arbitrary spin in de Sitter space (Λ ≠ 0) are well understood thanks to the seminal work of Deser & Nepomechie  and Deser & Waldron  - , in conjunction with that of Higuchi    .
(18) removes the scalar helicity-0 mode along the Higuchi partially-massless gauge line for Spin-2, leaving only 4 instead of 5 propagating degrees of freedom  —hence the term partially massless gravity. With respect to gravitational wave polarization analysis, this partially massless feature of EGΛ went unnoticed earlier on in initial polarization studies of gravitational waves which focused on Pauli-Fierz massive gravity effects    . The latter do not address partial masslessness in gravitational radiation behavior.
Derived directly from EGΛ in (1)-(3), (18) proves that EGΛ is a partially massless theory because that is specifically the Higuchi bound established by Deser and Nepomechie , Deser and Waldron  - , and articulated by Higuchi    . Massive gravity thus finds its roots when Einstein first introduced Λ into GR, rather than later when Pauli & Fierz (P-F)  pursued the study of massive gravity by adding appropriate terms to the Einstein-Hilbert Lagrangian.
Determining Λ from Gravitational Wave Observations: (18) is hence a direct prediction of EGΛ in (1). Recalling that gravitational wave observations can be used to determine the Hubble constant Ho  , we know that = Λ/3 in de Sitter space ( , Equation 2.6) from which Λ can be determined. Given the currently known disparity in Ho determinations  , Λ, mg, and Ho must eventually be brought into reconciliation. The question now becomes how to measure these effects using LIGO, VIRGO, and future LISA antenna configurations to determine whether polarization measurements can establish the loss of the helicity 0 excitation due to a scalar gauge symmetry but not the loss of helicity ±1, as predicted by the partially massless theory  .
In Conclusion: These results come directly from the RWZ Equation (9). The consequence is yet another way to determine the cosmological constant Λ, but from gravitational wave observations. It constitutes an entirely new prediction from Einstein’s theory, that Λ, c, Ho, and mg (having only 4 Spin-2 DOFs with helicities ±2, ±1), and conventional Λ-lore such as dark matter in ΛCDM models, are interrelated. For that reason alone, (18) needs to be verified experimentally. In addition, all of these parameters must collectively produce self-consistent values. The answer may also contribute to our understanding of galactic-rotation-curve behavior . Such predictions by EGΛ need to be investigated further.
The fundamental question for partially massive gravity is whether existing gravitational wave antenna configurations can be used to measure or determine the loss of the helicity 0 polarization caused by loss of a scalar gauge symmetry. It will probably require additional antenna configurations and possibly more antennas.
 Deser, S. and Nepomechie, R.I. (1983) Physics Letters B, 132, 321-324.
Deser, S. and Nepomechie, R.I. (1984) Annals of Physics, 154, 396-420.