Deriving the Properties of Space Time Using the Non-Compressible

Deriving the properties of space time using the non-compressible solutions of the Navier-Stokes Equations Ryan McDuffee∗ (Dated: September 24, 2019) Recent observations of gravitational waves by the Laser Interferometer Gravitational-Wave Ob- servatory (LIGO) has confirmed one of the last outstanding predictions in general relativity and in the process opened up a new frontier in astronomy and astrophysics. Additionally the observation of gravitational waves has also given us the data needed to deduce the physical properties of space time. Bredberg et al have shown in their 2011 paper titled From Navier-Stokes to Einstein, that for every solution of the incompressible Navier-Stokes equation in p + 1 dimensions, there is a uniquely associated dual” solution of the vacuum Einstein equations in p + 2 dimensions. The author shows that the physical properties of space time can be deduced using the recent measurements from the Laser Interferometer Gravitational-Wave Observatory and solutions from the incompressible Navier-Stokes equation. I. INTRODUCTION to calculate a hypothetical viscosity of space time and thus deduce the structure. Albert Einstein first published his theory of general Before moving forward with the analysis, it must be relativity in 1915, a year later he would predict the exis- understood that the Viscosity of space time that the au- tence of gravitational waves [1]. He based this prediction thor is intending to calculate, is an analogues property on his observation that the linearized weak field equa- rather than a literal one. Meaning that rather then try- tions had wave solutions [2][3], these transverse waves ing to estimate the thickness of a fluid like space time, we would be comprised of spatial strain and travel at the intend to show that by calculating the viscosity term in speed of light and would be generated by the time varia- a Navier-Stokes like equation, it is possible to calculate tions of the mass quadrupole moment of the source. Ein- a Shear modulus that describes the deformation of space stein also observed that the amplitude of these waves time, thus allowing for the analysis of space time at its would be incredible small and thus he never expected to most basic level. Before this can be done though, it is observe them [1]. That same year Karl Schwarzschild necessary to understand how the solutions of the Navier- would publish his solutions to the Einstein field equa- Stokes equation relate to the solutions of the Einstein tions that would predict the existence of black holes and field equation and how the data from the 2016 obser- in 1963 he would generalize these solutions to rotating vations at the Laser Interferometer Gravitational-Wave black holes[4][5]. Observatory (LIGO) can be used to calculate derive the In 2016 the Laser Interferometer Gravitational-Wave hypothetical properties of space time. Observatory(LIGO) announced they had detected a gravitational wave on September 14, 2015 at 09:50:45 UTC, The signals frequency ranged from 35 to 250 Hz with a II. THE NAVIER-STOKES EQUATION AND peak gravitational-wave strain of 1 ∗ 10−21[1]. This wave EINSTEIN FIELD EQUATION matched the waveform predicted by general relativity for the in spiral and merger of a pair of black holes and the Both the Einstein field equation Gµν = 0 and the 2 j ringdown of the resulting single black hole. Navier-Stokes equationv ˙i −η∂ vi +∂iP +v ∂j vi = 0 have arXiv:1607.05400v3 [gr-qc] 22 Sep 2019 The detection of such a wave is important on its own long been incredibly powerful tools for understanding the because the measurement of the waveform and amplitude world both terms of mathematics and physics. The Ein- of the gravitational waves from a black hole merger (such stein equation universally governs the long-distance be- as the one observed by LIGO) event makes possible ac- havior of gravitational systems, while the incompressible curate determination of its distance. The accumulation Navier-Stokes equation universally governs the hydrody- of black hole merger data from cosmologically distant namic limit of essentially any fluid. Both equations have events may help to create more precise models of the his- an intricate non-linear structure that allow them to be tory of the expansion of the universe and the nature of applied to wide verity of systems and thus offer us an the dark energy that influences it. However the direct insight into a wide range of physical phenomena. Be- detection of gravitational waves also allows use to deter- fore these two equation can be used to probe the struc- mine the physical properties of space, by using solutions ture of space time though it necessary to understand common to both the incompressible Navier-Stokes equa- that for that for every solution of the incompressible tion and the Einstein field equations of general relativity Navier-Stokes equation in p+ 1 dimensions, there is a uniquely associated dual” solution of the vacuum Ein- stein equations in p + 2 dimensions[6]. Those interested in a more rigorous demonstration of this assertion should ∗ rmcduff[email protected] read From Navier-Stokes to Einstein by Dr. Irene Bred- 2 berg et al[6]. obeys the linearized incompressible Navier-Stokes equa- i i 2 What follows is very brief overview their work, the tion ∂iv = 0, ∂τ v − η∂ vi = 0[6]. 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 ]+ ..

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