arXiv:1607.05400v3 [gr-qc] 22 Sep 2019 inadteEnti edeutoso eea relativity general of equations field Einstein equa- the Navier-Stokes solutions and incompressible tion using the by both space, to direct of deter- common to the properties use physical However of allows the nature also mine it. waves the gravitational influences and of that universe detection energy the his- dark of the expansion distantthe of the models cosmologically precise of more from tory create data to help merger accumulation may events The hole ac- possible black distance. makes its of event of LIGO) determination by (suchcurate observed merger one hole black the a as from waves gravitational amplitude and the waveform of the of the measurement the and because holes hole. black black of single pair resulting a the of of merger ringdown and for spiral relativity in general the by predicted waveform the matched h inl rqec agdfo 5t 5 zwt a with Hz 250 1 to of 35 UTC, strain from gravitational-wave 09:50:45 peak ranged at frequency 2015 signals 14, The September on wave grav- itational a detected had they announced Observatory(LIGO) rotating to solutions these generalize and holes[4][5]. equa- holes would black black field he of Einstein 1963 existence Schwarzschild the the in predict Karl to would year that solutions tions same his to That waves publish expected would these never [1]. he of thus them and amplitude observe small the incredible Ein- be that would source. observed the varia- of the also time moment the at stein quadrupole by mass travel generated the and be of would tions strain waves and spatial light transverse of of equa- these speed comprised field [2][3], be weak solutions would linearized wave the had that prediction tions this observation exis- based the his He predict on [1]. would waves he gravitational later of year tence a 1915, in relativity ∗ rmcduff[email protected] h eeto fsc aei motn nisown its on important is wave a such of detection The n21 h ae nefrmtrGravitational-Wave Interferometer Laser the 2016 In general of theory his published first Einstein Albert eiigtepoete fsaetm sn h non-compres the using time space of properties the Deriving httepyia rpriso pc iecnb eue usi deduced be can Observatory time equation. 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If we choose any structure of both the Einstein field equation and Navier- value for the viscosity other than η = rc, the constraint Stokes, and the boundary conditions necessary for there equations on c are still obeyed, but gravitational waves to be a solution to each equation. We begin by seek- are propagatedP down to I− and there is a singularity at ing a relation between the (p+2)-dimensional Einstein r = 0[6]. Dr. Bredberg solution goes one step further and and (p+1)-dimensional Navier-Stokes equations. Since, solves the problem in certain hydrodynamic and near- the former has a much larger solution space than the horizon limits without making a linearized approxima- latter, only a special type of Einstein geometry is rel- tion, enabling us to see a direct connection between the evant. Roughly speaking, the relevant geometries are nonlinear structures of the Navier-Stokes and Einstein non-singular perturbations of a horizon. A more pre- equations[6]. cises description of the geometries we are interested in We can now use a modified version of the Wilsonian can be found and From Navier-Stokes to Einstein by Dr. Approach to Fluid/Gravity in order to solve the shoot- Irene Bredberg et al [6] for the moment we are only in- ing problem in the long wavelength perturbation of c terested in the boundary conditions of these geometries without a simultaneous linearized expansion. The gen-P which we will denote as . The boundary hypersur- eral solution will be parametrized by ǫ of the full non- Pc face c is taken to be asymptotically null in both the linear Navier-Stokes equation with viscosity η = rc[6]. far futureP and far past. In Minkowskian coordinates First we must consider the metric ds2 = −rdτ 2 +2dτdr+ 2 i − i i i j i j dsp+2 = −dudv+dxidx , past null infinity I is the union dxidx −[2(1−r/rc)vidx dτ +(∂j v +∂iv )dx dx −2(r − 2 2 i of the null surface v → −∞ together with u → −∞ and r /2rc − rc/2)∂ vidx dτ] − δ(τ − τ∗)[(4(1 − r/rc)Fi + i i j c is a time like hyper surface uv = −4rc with v> 0[6]. 2/rcαi)dx dτ − 2/rcβij dx dx + ...][6], where Fi is an ar- P − + i Past (future) event horizons H H are defined by the bitrary function of xi obeying ∂iF = 0. βij and αi i boundaries of the causal future (past) of c near-horizon (which is divergence free) are both functions of x and P i and the hydrodynamic ǫ-expansion[6]. related to Fi by ∂ ∂j αi = Fi,βij = ∂iαj + ∂j αi[6]. Since These two expansions are shown to be equivalent η = the metric on is no longer flat, the constraint equa- Pc rc. Initial data can be specified on the union of c tions become linearized Navier-Stokes equations with − P i 2 i i and I . We consider initial data which is asymptoti- ∂τ v − η∂ v = F (x)δ(τ − τ∗). Since vi(x, τ) is taken to − cally Minkowskian and flat (no incoming waves) on I . vanish for τ > τ∗ the forcing term will cause it to jump i On c we generally demand that the intrinsic metric to Fi(x ) at τ = τ∗ after which it will evolve according to P i 2 i i γab be flat meaning that γab = ηabforab = 0, ...p. Next Navier-Stokes. Given v − η∂ v = F (x)δ(τ − τ∗)[6] this we wish to consider the general solution of the Einstein geometry solves the linearized Einstein equations every- equations consistent with this initial data and smooth on where, and is characterized by an arbitrary divergence- + H In particular. free vector field Fi(x). Before at τ = τ∗ it is flat while So far we have not specified the extrinsic curvature afterward it is, up to a coordinate transformation, the linearization of ds2 = −rdτ 2 + 2dτdr + dx dxi. At Kab on c or equivalently (and more conveniently) the p+2 i Brown-YorkP stress tensor on this will give us the the nonlinear level, the equations are cumbersome and Pc equation T ≡ 2γabK − Kab [6]. If no initial data were we have been unable to explicitly construct the analog of − 2 2 i i prescribed on I , any Tab on consistent with the con- ds = −rdτ +2dτdr + dxidx − [2(1 − r/rc)vidx dτ + c i j i j 2 2 i straint equations could be chosen.P This data could then (∂j v +∂iv )dx dx −2(r−r /2rc−rc/2)∂ vidx dτ]−δ(τ − i i j in general be evolved radially inwards to produce a space τ∗)[(4(1 − r/rc)Fi +2/rcαi)dx dτ − 2/rcβij dx dx ]+ ... time everywhere inside of [6].In general, such a space away from [6]. However it seems plausible that qual- c Pc time will have gravitationalP flux (if not singularities) go- itatively similar solutions persist at the nonlinear level. ing up to v = ∞(I+) as well as down to I−.Hence we have a shooting problem” to find those special allowed + choices of Tab which produce a space time smooth on H III. THE 2016 OBSERVATION OF with no flux coming from I+[6]. GRAVITATIONAL WAVES Dr. Bredberg has devised a complete solution to this problem in [7]. We then implement Dr. Bredberg so- On September 14th 2015, the Laser Interferometer lution to leading order in a double expansion in long Gravitational-Wave Observatory (LIGO) directly ob- wavelengths and weak fields. Ingoing Rindler coordi- served gravitational waves from the inward spiral and nates were used for which the leading order at metric merger of a pair of black holes of around 36 and 29 solar 2 2 i is dsp+2 = −rdτ +2dτdr + dxidx c is the acceler- masses [1]. Up until then existence of gravitational waves − P + ated surfacer = rc, H , is τ = −∞ and H is r=0[6]. had only been inferred indirectly, through their effect on These coordinates are convenient for analyzing smooth- the timing of pulsars in binary star systems [8]. The + ness on H . It was found that the allowed choices of Tab observation of these waves gives us the information we are precisely those corresponding to the linearized fluid: need to use the Navier-Stokes equation to calculate the 3/2 τi 3/2 ij ij rc T = vi, rc T = −η∂ [6]. Where the (kinematic) hypothetical viscosity of space time, primarily it allows viscosity here is given by the formula η = rc while vi us to calculate the frequency and amplitude of the gravi- 3 tational wave. This calculation was possible prior to the terested in the field far from a source, we can treat the detection of Gravitational waves, however with the di- source as a point and everywhere else, the stressenergy αβ rect observation of the waves, we can now use data that tensor would be zero, so our equation becomes h = 0. is physically meaningful. Now, this is just the usual homogeneous wave equation The two black holes involved in the merger had masses αβ for each component of h . For a wave moving away of approximately 36(+5, -4) and 29(+ or 4) solar from a point source, the radiated part (meaning the part masses[1]. The recorded power of the gravitational wave 4 that dies off as 1/r far from the source) can always be peaked at 3.6 ∗ 10 9 watts and Across the 0.2-second du- f(t−r,θ,φ) ration of the detectable signal, the relative tangential (or- written in the form r , where f is just a generic biting) velocity of the black holes increased from 30 function. It can be shown that it is always possible to We begin with the equation for the space time curva- make the field traceless[11]. ture expressed with respect to a covariant derivative ▽, Now, if we further assume that the source is posi- tioned at r=0, the general solution to the wave equation in the form of the Einstein tensor Gµν . We can then re- in spherical coordinates is late the curvature to the stressenergy tensor Tµν , with 8πGn the equation G = 4 T [9]. Where G is Newton’s µν c µν n 00 0 0 gravitational constant, and c is the speed of light, for αβ 1 00 0 0  the purpose of this paper we will use geometrized units h = 0 0 h (t − r, θ, φ) h (t − r, θ, φ) so that G =1= c in order to simplify our calcula- r  + ×  n 0 0 h (t − r, θ, φ) h (t − r, θ, φ) tions. We can now rewrite the Einstein’s equations as × + wave equations and define our flat space, our flat space [11]. Thus we have our polarization equations which time will be given as can now be rewritten in Newtonian terms as h+ = 2 −1 G 4m1m2 −100 0 R c4 r [9] and to calculate the amplitude of the  010 0  gravitational wave detected by the Laser Interferome- η = µν  0 0 r2 0  ter Gravitational-Wave Observatory we simply substitute  2   0 00 r2 sin θ the measured values and get an upper value of 25956/R or a lower value of 19890/R, where R is the distance from (it should be noted that in the next section η[9] will also the center of the system. The collision was detected ap- be used to calculate our hypothetical viscosity of space proximately 1.3 billion Light years away from earth [1], time.) This flat-space metric has no physical significance; and thus the spatial distortion caused by the gravita- it is a purely mathematical device necessary for the anal- tional wave was between 2.1∗10−21 meters and 1.6∗10−21 ysis. meters. This is the amplitude of the gravitational wave Now, we can also think of the physical metric gµν as detected by the Laser Interferometer Gravitational-Wave a matrix and find its determinant det g. Finally we de- Observatory and with the amplitude of the wave, we can fine our radiation field in terms of our flat space time now calculate the hypothetical properties of space time gµν and find the determinant det g, this will give us by using the Navier-Stokes equation. αβ h ≡ ηαβ − gαβ detg . Next we define out coordi- q nates in such a way that this quantity satisfies the de IV. THE THEORETICAL PROPERTIES OF Donder gauge condition (conditions on the coordinates): SPACE TIME AS FLUID αβ ▽βh = 0. Where ▽ represents the flat-space deriva- tive operator. These equations say that the divergence of We have now established that for that for every solu- the field is zero. The linear Einstein equations can now tion of the incompressible Navier-Stokes equation in p+ 1  αβ αβ  2 be written as h = 16πτ [10] where = −∂t + △ dimensions, there is a uniquely associated dual” solution is the flat-space d’Alembertian operator, and τ αβ repre- of the vacuum Einstein equations in p + 2 dimensions [6] sents the stressenergy tensor plus quadratic terms involv- and calculated the amplitude of the wave 2016 gravita- αβ ing h [10].This is just a wave equation for the field with tional wave. We also know that the gravitational wave a source, despite the fact that the source involves terms detected by The Laser Interferometer Gravitational- quadratic in the field itself. That is, it can be shown Wave Observatory has velocity of c[1]. With this infor- that solutions to this equation are waves traveling with mation we can now use the Navier-Stokes equation to velocity 1 in these coordinates. derive an equation for the velocity of wave in an incom- In order to obtain a numerical result from this equa- pressible medium and calculate the Shear modulus. We tion though it will need to be linearized. We assume begin my considering a wave as disturbance of a fluid-like that space is nearly flat, so the metric is nearly equal to medium[12][13][14]. the ηαβ tensor. This means that we can neglect terms Next we write out the equations for the conservation ∂ρui ∂ρ ∂ρui αβ αβ of mass and momentum as + = 0()and uj + quadratic in h , which means that the τ field reduces ∂xi ∂t ∂xj αβ ∂ρui ∂σij to the usual stressenergy tensor T and Einstein’s equa- ∂t − ∂x = 0[12][13][14] where ρ is the mass density αβ j tions become h = 16πT αβ.However since we are in- (for our calculations this will be the smallest possible 4

Fl deformation in space time) u is the particle velocity, P is the shear modulus for space time G = A△x where F our fluid pressure. It should again be noted that when is the force of the two observed black holes colliding, l we refer to things like mass, particle and pressures, we is the initial length of the smallest area effected by the are speaking metaphorically about analogous variables force, A is the area over which the force acts (this would in continuum mechanics. The physical meaning of each be the circular distance between earth and the colliding of the analogous variables will be covered in the next black holes) and △x would be the spatial displacement section. We also have our three spatial coordinates are caused by the gravitational wave (the amplitude of the xi(i =1, 2, 3) for the domain ω. Particle velocities u are gravitational wave we calculated earlier). for each direction xi. Using the Navier-Stokes equation We now know all of the quantities for our velocity equa- we can find the equation for incompressible fluid flow tion except for the initial length of the smallest area af- du ρ dt = fb −▽P and the body forces are negligible giving fected by the wave. By substituting the known quantities us fb = 0[12]. into the wave velocity equation and solving for l, we find The force f(t) acts as a source function upon the that the smallest area affected by the observed gravita- domainω. Any force acting within the domain causes tional wave was 1.4∗10−21 meters. We can now take this pressure and density changes, and the fluid nature of information and substitute our results into the equation the medium will create an equilibrium restoring force[15]. u for dynamic viscosity F = µA y and solve for µ to get Next we consider small perturbations △ in the density ρ, our hypothetical viscosity of space time of 6.2616 ∗ 10−39 the particle velocity u, and the pressure P from the ini- kg/ms. This for all intents and purposes gives us a sub- tial rest conditions which are labeled with subscript 0. stance with a viscosity that is functionally 0, and thus ut = u0 + △u, ρt = ρ0 + △ρ, Pt = P0 + △P [15]. The ini- space time behaves as a perfect fluid. tial particle velocity of our location in space is assumed to be 0 because the domain ω is assumed to be at rest relative to the particles around it. The density perturba- V. IMPLICATIONS tion is based on the acoustic approximation[15], in this approximation A fluid has a pressure which is a function Before continuing to the consequences of space time of density, temperature, and gravitational forces. behavior as a perfect/super fluid, let us first summarize We shall assume that the gravity forces are relatively our argument. We have already established that for ev- constant over the domain ω and do not exert any differ- ery solution of the incompressible Navier-Stokes equation ential force on the fluid. We will neglect the effects of in p + 1 dimensions, there is a uniquely associated dual” temperature as the changes are very small. Hence, we solution of the vacuum Einstein equations in p + 2 di- will assume that only the density is important and that mensions. This means that any solution derived from the stress within the fluid is related to the strain as a Navier-Stokes equation in p + 1 dimensions there is a function of density. Next we simply apply the Kronecker duel solution of the vacuum Einstein equations in p + 2 delta to our stress matrix dimensions. This establishes that our work in one set of 1 0 0 equations is valid in the other assuming that both treat   δij = 0 1 0 their respective mediums as incompressible and that a 0 0 1 constant number of dimensions is used in each. We then take the polarization equation for a system du of orbiting bodies and calculate the amplitude of the and use Eulers equation to get ρ dt = −▽Pt and since the pressure is assumed to be constant the equation becomes gravitational wave caused by this system using the val- du ues measured by the Laser Interferometer Gravitational- ρ dt = −▽△P [16]. Now the initial medium is at rest and has no convective acceleration, which permits changing Wave Observatory in 2015 and then using a shear mod- ∂u ule and the equation for the velocity of a transvers wave the form of the derivatives and this gives us ρ ∂t = −▽ △P . Next u is the gradient of φ, and we can also see in a medium derived from the Navier-Stokes equation that the product of △ρ and the gradient of φ will be and then solve for the initial length of the space prior to ∂▽φ its distortion. We then take the force of the event that small thus ▽φ = u and ρ0 ∂t = −▽△P [12]. Next we assume the derivatives of time and space can caused the spatial distortion, the initial size of the undis- ∂φ torted space, the area affected by the gravitational wave be exchanged which will give us ρ0 = −▽△P and this ∂t and then solve for the viscosity of space. The resulting equation can be simplified to ρ ∂φ = −△ P . Next since 0 ∂t viscosity tells us about how malleable space time is and we are measuring space time in terms of its deformation as a result imposes constraints on the size of any extra di- caused by waves, we use the shear modulus instead of mensions space. In this context ”viscosity” simply means the Young’s modulus that would normally be used for the deformability of space time, our particles are simply calculating the velocity of a wave. To calculate the wave quantized units of space, and the ”pressure” is a uniform velocity, we take the incompressible fluid equation and force present through out all of space time. apply mass conservation to the first derivative of the fluid According to general relativity, the conventional gravi- equation and then solve for velocity which gives us u = tational wave is the small fluctuation of curved space time G (where G is the shear modulus). Next we calculate q ρ0 which has been separated from its source and propagates 5 independently. In Superfluid vacuum theory, a subset our first problem with trying to deduce the size of these of Grand Unified Field Theories where the fundamental extra dimensions via the characteristics of gravitational physical vacuum (non-removable background) space time wave. If we simply take the general case of the EinsteinS- is viewed as a Superfluid and the curved space time we moluchowski relation and calculate the mobility µ of our see in general relativity is the small collective excitation particles of space for the velocity of the expanding uni- of a superfluid background[17]. Meaning that much like verse at a given distance (calculated using Hubble’s law) water, space time is composed of smaller parts. However over the force that caused the gravitational wave in con- the properties of these parts will vary widely depending junction with the ambient temperature of the vacuum, we on what formulation of Superfluid vacuum theory is be- can then solve for the diffusion constant, the above values ing used and if Superfluid vacuum theory is being incor- give space time a diffusion constant of 8.3097∗10−57m2/s. porated into another theory such as m-theory. However when treating space time as a super fluid and assum- We then take the StokesEinstein equation[19] and set ing that general relativity’s interpretation of a gravita- it equal to the diffusion constant that we just calculated tional wave is accurate, results in the following conclu- and then substitute our viscosity for n and then sim- sions about the graviton. Primarily, the Graviton would ply solve for r. If we do this then we get a radius of be the ”small fluctuation of the small fluctuation” mean- 3.799 ∗ 1070m to put this in perspective consider that ing it would be the fundamental unit of space. This may the radius of the observable universe is 4.4 ∗ 1026m. This not appear to be a physically robust concept as it is akin would seem to indicate that any extra dimension of space to trying to define a phonon in terms of smaller fluctu- would have to be large extra dimensions akin to those in ations inside it. However it may be possible to better the RandallSundrum model[20]. However we initially in- ascertain the nature of such a strange phenomenon if we tended to calculate the size of compactified dimensions make the following assumptions about space times struc- such as those in m-theory. However in order to do this we ture as a fluid. need to calculate the drift velocity of the individual units First we assume that there are more than 3+1 dimen- that in theory would comprise space time, however at this sion of space and that any extra dimensions of space level, determining the exact the velocity of any individ- can exist both at the quantum level and on the macro- ual unit becomes impossible due to the laws of quantum scopic level. Next we assume there are differences in the physics. Thus while our calculations on the macroscopic malleability of these dimensions of space time, meaning level introduce the intriguing possibility of large extra that higher dimensions are less malleable then the dimen- dimensions of space, in order to use the data measured sions we experience every day. With these assumption by the Laser Interferometer Gravitational-Wave Observa- in place, in theory we should be able to calculate the tory to deduce the nature of compactified extra dimen- size of these extra dimension using the EinsteinSmolu- sions of space, a more complete description of space time chowski relation[18]. However this is where we encounter at the quantum level is required.

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