_{1}

^{*}

It is shown in Einstein gravity that the cosmological constant
Λ introduces a graviton mass
*m*
_{g} into the theory, a result that will be derived from the Regge-Wheeler-Zerilli problem for a particle falling onto a Kottler-Schwarzschild mass with
Λ ≠ 0. The value of mg is precisely the Spin-2 gauge line appearing on the
Λ -
<span style="white-space:nowrap;">m<sup>2</sup><sub style="margin-left:-6px;">g</sub></span> phase diagram for Spin-2, the partially massless gauge lines introduced by Deser & Waldron in the
<span style="white-space:nowrap;">m<sup>2</sup><sub style="margin-left:-6px;">g</sub></span>,
Λ) phase plane and described as the Higuchi bound
<span style="white-space:nowrap;">m<sup>2</sup><sub style="margin-left:-6px;">g</sub></span>= 2
Λ/3. Note that this graviton is unitary with only four polarization degrees of freedom (helicities ±2, ±1, but not 0 because a scalar gauge symmetry removes it). The conclusion is drawn that Einstein gravity (EG,
Λ ≠ 0) is a partially massless gravitation theory which has lost its helicity 0 due to a scalar gauge symmetry. That poses a challenge for gravitational wave antennas as to whether they can measure the loss of this gauge symmetry. Also, given the recent results measuring the Hubble constant
*H*
_{o} from LIGO-Virgo data, it is then shown that
Λ can be determined from the LIGO results for the graviton mass
*m*
_{g} and
* H*
_{o}. This is yet another multi-messenger source for determining the three parameters
Λ,
* m*
_{g}, and
*H*
_{o} in astrophysics and cosmology, at a time when there is much disparity in measurements of
*H*
_{o}.

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 [

One begins with a small perturbative expansion of the Einstein field equations

R μ ν − 1 2 g μ ν R + Λ g μ ν = − κ T μ ν (1)

about the known exact solution η_{μν} where the metric tensor is g μ ν = η μ ν + h μ ν , with h_{μν} the dynamic perturbation of the background raising and lowering operator η_{μν}. The most general spherically symmetric solution is well-known to be a Kottler-Schwarzschild (KS) metric

d s 2 = − e ν d t 2 + e ζ d r 2 + r 2 d Ω 2 (2)

where

e ν = 1 − 2 M r − Λ 3 r 2 = e − ζ (3)

with M = G M * / c 2 , d Ω 2 = ( d Θ + sin 2 d ϕ 2 ) , and η μ ν = d i a g ( e ν , e − ν , r 2 , r 2 sin 2 Θ ) in spherically symmetric coordinates. Its contravariant inverse η^{μν} is defined such that η μ ν η μ ν = δ μ ν .

The wave equation for gravitational radiation h_{μν} 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 η μ ν = g ( 0 ) μ ν produces the following

[ h μ ν ; α ; α − h μ α ; ν ; α − h ν α ; μ ; α + h α α ; μ ; ν ] + η μ ν [ h α γ ; α ; γ − h α α ; γ ; γ ] + h μ ν ( R − 2 Λ ) − η μ ν h α β R α β = − 2 κ δ T μ ν (4)

Stability must be assumed in order that δT_{μν} is small. This equation can be simplified by defining the function (introduced by Einstein himself)

h ¯ μ ν ≡ h μ ν − 1 2 η μ ν h (5)

and its divergence

f μ ≡ h ¯ μ ν ; ν (6)

Substituting (5) and (6) into (4) and re-grouping terms gives

h ¯ μ ν ; α ; α − ( f μ ; ν + f ν ; μ ) + η μ ν f α ; α − 2 h ¯ α β R α μ 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 (f_{μ} = 0), and suppresses any vector gravitons. Wave Equation (7) reduces to

h ¯ μ ν ; α ; α − 2 h ¯ α β R α μ ν β − h ¯ μ α R α ν − h ¯ ν α R α μ − η μ ν h α β R α β + h μ ν ( R − 2 Λ ) = − 2 κ δ T μ ν (8)

In an empty (T_{μν} = 0), Ricci-flat (R_{μν} = 0) space without Λ (R = 4Λ = 0), (8) further reduces to

h ¯ μ ν ; α ; α − 2 R α μ ν β h ¯ α β = − 2 κ δ T μ ν (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 η_{μν} 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 4 Λ − R = − κ T , whereby they become

R μ ν − Λ g μ ν = − κ [ T μ ν − 1 2 g μ ν T ] (10)

For an empty space (T_{μν} = 0 and T = 0), (10) reduces to de Sitter space

R μ ν = Λ g μ ν (11)

and the trace to R = 4Λ.

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

h ¯ μ ν ; α ; α − 2 R α μ ν β h ¯ α β = − 2 κ δ T μ ν (12)

One can arrive at (12) to first order in h_{μν} by using g_{μν} as a raising and lowering operator rather than the background η_{μν}—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 R^{α}_{μν}^{β} 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 η_{μν} to zero, produces the de Sitter space (11) of constant curvature K = 1/R^{2}, where we can focus on the effect of Λ. The Riemann tensor is now

R γ μ ν δ = + K ( g γ ν g μ δ − g γ δ g μ ν ) (13)

and reverts to

R α μ ν β = + K ( g α ν g μ β − g α β g μ ν ) (14)

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

− 2 K [ ( h ¯ μ ν − η μ ν h ¯ ) + ( h ¯ α μ h α ν + h ¯ ν β h β μ − h ¯ h μ ν − η μ ν h α β h ¯ α β ) ] + [ 2 h μ α h ¯ α ν + η μ ν h α β 2 ] (15)

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

h ¯ μ ν ; α ; α − 2 3 M h ¯ μ ν + 2 3 Λ η μ ν h ¯ = − 2 κ δ T μ ν (16)

There is no cancellation of the Λ contributions to first order. Noting from (5) that h ¯ = h ( 1 − 1 / 2 η ) , then a traceless gauge h ¯ = 0 means either that h = 0 or η = 2. Since η = 4, (16) reduces to

h ¯ μ ν ; α ; α − 2 3 Λ h ¯ μ ν = − 2 κ δ T μ ν (17)

in a traceless Hilbert-Einstein-de Donder gauge where h ¯ μ ν ; ν = 0 and h ¯ μ μ = 0 . (17) is a wave equation involving the Laplace-Beltrami operator term h ¯ μ ν ; α ; α for the Spin-2 gravitational perturbation h ¯ μ ν bearing a mass

m g = 2 Λ / 3 (18)

similar to the Klein-Gordon Equation ( □ − m 2 ) φ = 0 for a Spin-0 scalar field φ in flat Minkowski space. The Locally Flat Limit section which follows demonstrates that h ¯ μ ν ; α ; α → □ h ¯ μ ν in (17) for the limit r → 0 . From (17) and (18) then

( □ − m g 2 ) h ¯ μ ν = − 2 κ δ T μ ν (19)

in the locally flat-space limit r ≪ 1 .

Note that Penrose [

m g = Λ / 3 (20)

which is the surface gravity κ_{C} = m_{g} of the cosmological event horizon identified by Gibbons & Hawking [

Locally Flat Limit of Wave Equation (17): It is necessary to demonstrate that hidden Λ-terms arising from h ¯ μ ν ; α ; α in (17) do not cancel the mass term in (18)-(20) when r → 0 and h ¯ μ ν ; α ; α → h ¯ μ ν , α , α = □ h ¯ μ ν , 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 r^{2}dΩ^{2} in (2) is of second-order in r and is negligible as r → 0 . Thus the focus is on e^{ν} (with M = 0) in (3) appearing in the diagonal of η_{μν} and its inverse η^{μν}. Hence, η_{00} = −c and η^{00} = −c^{−1}, while η_{11} = c^{−1} and η^{11} = c. Also, note that c ( r ) → 1 and c ( r ) − 1 → 1 as r → 0 .

Introducing the Christoffel symbol Γ α β γ , we can write

h ¯ μ ν ; α ; α = g α β h ¯ μ ν ; α ; β = g α β [ h ¯ μ ν , α ; β − ( Γ α μ ε h ¯ ε ν ) ; β − ( Γ α ν ε h ¯ μ ε ) ; β ] (21)

Define

h ¯ μ ν ; α ; α = □ h ¯ μ ν + A μ ν + B μ ν + C μ ν (22)

where

□ h ¯ μ ν = h ¯ μ ν , α , α (23)

A μ ν = − Γ β μ ε h ¯ ε ν , β − Γ β ν ε h ¯ μ ε , β − Γ β α ε h ¯ μ ν , ε η α β − Γ α μ ε h ¯ ε ν , α − Γ α ν ε h ¯ μ ε , α (24)

B μ ν = − ( Γ α μ ε ) , α h ¯ ε ν − ( Γ α ν ε ) , α h ¯ μ ε (25)

C μ ν = − η α β [ ( Γ β δ ε Γ α μ δ − Γ β α δ Γ δ μ ε − Γ β μ δ Γ α δ ε ) h ¯ ε ν − Γ β ε δ Γ α μ ε h ¯ δ ν − Γ β ν δ Γ α μ ε h ¯ ε δ + ( Γ β δ ε Γ α ν δ − Γ β α δ Γ δ ν ε − Γ β ν δ Γ α δ ε ) h ¯ μ ε − Γ β μ δ Γ α ν ε h ¯ δ ε − Γ β ε δ Γ α ν ε h ¯ μ δ ] . (26)

B_{μν} is the term of interest. A_{μν} and C_{μν} contain factors of second order, or terms that vanish in locally flat space ( r ≪ 1 ). Furthermore, only the first-order second derivatives in B_{μν} remain as r → 0 . These terms are

B α μ ν ∗ α = − 1 2 η ε γ [ ( η α γ , μ , α + η μ γ , α , α − η α μ , γ , α ) h ¯ ε ν + ( η α γ , ν , α + η ν γ , α , α − η α ν , γ , α ) h ¯ μ ε ] (27)

which can be defined as

B α μ ν ∗ α = F μ ν + G μ ν + H μ ν (28)

where

F μ ν = − 1 2 η ε γ [ ( □ η μ γ ) h ¯ ε ν + ( □ η ν γ ) h ¯ μ ε ] (29)

G μ ν = − 1 2 η ε γ [ η α γ , μ , α h ¯ ε ν + η α γ , ν , α h ¯ μ ε ] (30)

H μ ν = + 1 2 η ε γ [ η α μ , γ , α h ¯ ε ν + η α ν , γ , α h ¯ μ ε ] (31)

In this approximation, □ = − ∂ t 2 + ∇ 2 → ∇ 2 . Also □ η 00 → ∇ 2 η 00 = + 2 / 3 λ and □ η 11 → ∇ 2 η 11 = + 2 / 3 λ .

We find that

F μ ν = − 1 2 η 00 [ ( □ η μ 0 ) h ¯ 0 ν + ( □ η ν 0 ) h ¯ μ 0 ] − 1 2 η 11 [ ( □ η μ 1 ) h ¯ 1 ν + ( □ η ν 1 ) h ¯ μ 1 ] (32)

whereby (all other terms do not contribute)

F 00 = − η 00 [ ( □ η 00 ) h ¯ 00 ] = + 2 3 λ h ¯ 00 (33)

F 11 = − η 11 [ ( □ η 11 ) h ¯ 11 ] = − 2 3 λ h ¯ 11 (34)

Next

G μ ν = − 1 2 η 11 [ η 11 , μ , 1 h ¯ 1 ν + η 11 , ν , 1 h ¯ μ 1 ] (35)

whereby (all other terms do not contribute)

G 01 = − 1 3 λ h ¯ 01 ; G 10 = − 1 3 λ h ¯ 10 ; G 11 = − 2 3 λ h ¯ 11 . (36)

And lastly,

H μ ν = 1 2 η 11 [ η α μ , 1 , α h ¯ 1 ν + η α ν , 1 , α h ¯ μ 1 ] (37)

whereby

H 00 = 0 ; H 11 = 2 3 λ h ¯ 11 ; H 01 = 1 3 λ h ¯ 01 ; H 10 = 1 3 λ h ¯ 10 (38)

Summarizing, the two contributing terms to F_{μν} in (33) and (34) are equal and opposite thereby cancelling in (32). Thus, F_{μν} = 0. Similarly, the collective G_{μν} and H_{μν} terms in (36) and (38) cancel one another, giving G_{μν} + H_{μν} = 0. Hence B α μ ν ∗ α = B μ ν ≡ 0 in (28) and (25). Therefore we get h ¯ μ ν ; α ; α → h ¯ μ ν , α , α = □ h ¯ μ ν 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 [

(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 [

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 [

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 H_{o} [_{o} determinations [_{g}, and H_{o} 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, H_{o}, and m_{g} (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.

The author declares no conflicts of interest regarding the publication of this paper.

Wilson, T.L. (2020) Determining the Cosmological Constant Using Gravitational Wave Observations. Journal of Modern Physics, 11, 1-8. https://doi.org/10.4236/jmp.2020.111001