The Schwarzschild Solution to the Nexus Graviton field Stuart Marongwe Physics Department, McConnell College, Tutume, Botswana stuartmarongwe@gmail.com The Schwarzschild approach is applied to solve the field equations describing a Nexus graviton field.The resulting solutions are free from singularities which have been a problem in General Relativity since its inception.Findings from this work also demonstrate that at the Hubble radius, the metric signature of space-time changes generating short lived but intense bursts of energy during the transition process.The solutions in this paper also provide an explanation to the enigma of late time cosmic acceleration, the galaxy rotation curve problem and the coincidence problem. Keywords: Quantum Gravity; General Relativity; Schwarzschild Solution;Dark Energy;Dark Matter Mathematics Subject Classification 2010: 83C27, 83C45, 85A40 1. Introduction Modern physical cosmology is modelled upon the highly successful theory of General Relativity (GR) which has so far passed many unambiguous observational and experimental tests.Indeed the publication of the final version of GR in 1917 provided a sound scientific base for cosmology and helped establish it on an equal footing as other scientific disciplines. As cosmology continues to study the origin, evolution, and eventual fate of the universe, it seeks a quantum version of GR which will provide a more comprehensive description of cosmic phenomena. Phenomena such as Dark Matter(DM) and Dark Energy (DE) continue to pose a serious challenge to our understanding of nature.There is a general consensus among those who study these enigmatic phenomena that Quantum Gravity might clear these challenges[1,2]. Recently, a paper [3] was advanced inwhich a plausible self-consistent quantum version of GR was proposed. In the Nexus paradigm, space-time is seen as a graviton field with 1060 eigen states.The Nexus graviton itself is a perturbation of this graviton field or a pulse of space-time. f (xµ ) = ( µ ∆rHS µ if |xµ | < rHS 0 µ if |xµ | > rHS 1 (1) 2 Stuart Marongwe where rHS = c/H0 is the present Hubble radius and H0 the present Hubble constant in S.I units(H0 = 2, 30 × 10−18 s−1 ) as determined by the WMAP Collaboration over a period of nine years of data capture [4]. The pulse can be expressed as a space-time wave packet through the application of fourier expansion. Z k µ µ cos(kµ xµ ) sin(kµ rHS )dkµ 2γµ ∆rHS (2) f (xµk ) = π k µ 0 Here the spin 2 nature of the graviton arises from the summation of the four spin half components of a displacement vector in four space. The graviton energy is given by the expression ∆En = nhH0 (3) and the cosmological constant Λ is calculated as  2 2  E0 H0 =3 = 1.76 × 10−52 m−2 Λ=3 c hc (4) The four-momentum associated with the presence of a graviton in the nth quantum state gives rise to a stress momentum tensor as described in Einstein’s field equations causing a rotational distortion to the four vector. 1 ′ 8πG ′ ′ R(k)µν − Rg(k)µν = n2 4 ρvac g(k 0 )µν 2 c (5) where ρvac c4 3c4 = Λ= 8πG 8πG  E0 hc 2 (6) ′ ′ Here R(k)µν is the Ricci curvature tensor on a modified metric of g(k)µν for a given eigen state of space-time generated by a graviton field with a range of four wave vectors up to kµ , c the speed of light and G is the Newtonian gravitational constant. Thus the Nexus graviton can be seen as a FLRW fluid of spherical symmetry with a characteristic radius r = rHS n .This fluid is in the form of vacuum energy which has a negative pressure and thus causes the Nexus graviton (wave packet) to expand.The Nexus graviton thus generates both gravity by curving space-time and also a negative vacuum pressure.From the paper [3], we find that the graviton expands by transiting into a lower energy state which is accompanied by the emission of low energy graviton E0 = hH0 . If this process is taken into consideration then equation(5) can be written as 1 ′ 8πG ′ ′ R(n−1)k − Rg(n−1)k = (n2 − 1) 4 ρvac g(k 0 µν 0 µν 0 )µν 2 c (7) The findings from [3] also show that a graviton in the nth quantum state induces a uniform rotational speed v = H0 r = 2πH0 kn (8) The Schwarzschild Solution for the Nexus Graviton 3 and a centripetal acceleration of a = H0 c/n = H0 v (9) We seek in this paper, to apply the Schwarzschild solution to the field equations (5) and investigate the resulting solutions. 2. The Schwarzschild solution Following Karl Schwarzschild (see [5,6]for a detailed derivation), we solve the field equations describing a Nexus graviton to find the space-time geometry associated stationary, spherical distribution of vacuum matter of mass Z V 8πG dV. (10) Mvac = n2 4 ρvac c Since from equation(1) the space outside the event is empty, the energy-momentum tensor Tµν vanishes, so the field equation becomes: 1 ′ ′ =0 R(k)µν − Rg(k)µν 2 The appropriate boundary conditions are: (11) • metric must match interior metric at the bodys surface; • metric must assume the form of a flat (Minkowski) metric far away from the body. This is the Schwarzschild problem which is solved by solving for the Schwarzschild ′ metric gµν starting from a consideration of a general static and isotropic metric: • static: both time-independent and symmetric under time reversal ; • isotropic: invariant under spatial rotations . The interval satisfying these criteria may be written as: ds2 = A(r)dt2 + B(r)dr2 + r2 dθ2 + sin2 θdφ2 (12) where the first two terms on the RHS describe radial behavior (isotropy), and the last two the surface of the sphere (spherical symmetry). It can be expressed in many equivalent forms. One convenient form is: ds2 = eN (r) dt2 + eP (r) dr2 + r2 (dθ2 + sin2 θdφ2 ) (13) The Schwarzschild problem is now reduced to solving for N (r) and P (r) from the field equations and the appropriate boundary conditions.Following Schwarzschild, the solution to the Schwarzschild problem is 2GMvac 2 2 2GMvac −1 2 )c dt + (1 − ) dr + r2 (dθ2 + sin2 θdφ2 ) 2 c r c2 r Given that v = H0 r and that GMvac = v 2 = (H0 r)2 r ds2 = −(1 − (14) (15) 4 Stuart Marongwe then the Schwarzschild solution for the Nexus graviton reduces to ds2 = −(1 − 2r2 2r2 2 2 )c dt + (1 − 2 )−1 dr2 + r2 (dθ2 + sin2 θdφ2 ) 2 rHS rHS (16) 2 Also given that n2 = rRS /r2 then equation (16) can be written as ds2 = −(1 − 2 2 2 2 )c dt + (1 − 2 )−1 dr2 + r2 (dθ2 + sin2 θdφ2 ) n2 n (17) 3. Physical intepretation of the solutions One prominent feature of equation (17) is that at the Planck state (n = 1060 ) the metric reduces to a flat Minkowski metric. ds2 = −c2 dt2 + dr2 + r2 (dθ2 + sin2 θdφ2 ) (18) At these high energy states and on a flat Minkowski metric,quantum phenomena reign supreme.However the most remarkable feature of the same equation is the lack of a singularity which is present in classical GR at the quantum state n = 1 i.e at the Hubble radius. In this quantum state the metric undergoes a signature change and the line element becomes ds2 = c2 dt2 − dr2 + r2 (dθ2 + sin2 θdφ2 ) (19) Metric signature changes are accompanied by the production of particles and this phenomenon has been studied extensively by [7-9].The characteristic feature of this phase transition appears to be a kind of big bang due to the sudden emergence of near infinite particle density of states at the vibrational frequency modes of the space-time graviton field ωn .The density of states of the particles at these frequencies diverges in the lossless space-time continuous medium limit: Kmax ω dn = (20) dω π 2 c2 Where Kmax is the momentum cutoff for the lossless space-time medium.In toy models using metamaterials, this phenomenon manifests itself as short bursts of intense electromagnetic radiation.Recently Fast Radio Bursts similar to these have been observed by radio telescopes [10,11] coming from different cosmic sources.Another source could be a metric transition process via some unknown astrophysical process which can cause space-time to transit into the n = 1 quantum state. 3.1. Dependence of Frequency Redshifts and Time Dilation with Radial distances Gravitons of large four wave vector have infinitesimal line elements in line with the Uncertainity Principle.This results in an equally infinitesimal proper motion v = 2πH0 /kn of a test particle placed at a radial distance r = 2π/kn within the The Schwarzschild Solution for the Nexus Graviton 5 graviton, compared to a test particle placed in gravitons of small four wave vector.Thus at radii approaching the Planck length, motion in the classical sense is imperceptable.A comparison of relative motion at different radii can be made by comparing the redshift of light emitted by an atom within a graviton of radius r +D to that emitted by an atom within a graviton of radius r.Applying the Schwarzschild solution for gravitational redshift and time dilation to the Nexus paradigm , the wavelength of light at r, λr compared to the wavelength at r + D, λr+D is λr+D v u 2 u 1 − 2r 2 rHS u = λr t 2 1 − 2(r+D) r2 (21) HS Time dilatation occurs at large radii of the graviton compared to small radii and is the cause of the redshift.The time transitions at (r + D) compared to those at r is therefore v u 2 u 1 − 2r 2 rHS u (22) τr+D = τr t 2 1 − 2(r+D) 2 r HS 4. The Extended Solution We now seek to include the contribution to the perturbation of a spacetime graviton field by a graviton associated with baryonic matter of mass Mbar . This can be included into equation(7) which then becomes 8πG 1 ′ ′ ′ = 4 (T(k)µν + (n2 − 1)ρvac g(k R(k+(n−1)k − Rg(k+(n−1)k ) 0 )µν 0 )µν 0 )µν 2 c (23) Therefore the Schwarzschild solution for this case becomes 2G(Mbar + Mvac ) 2GMΛ r 2 2 2G(Mbar + Mvac ) 2GMΛ r −1 2 2 2 + 2 2 )c dt +(1− + 2 2 ) dr +r (dθ +sin2 θdφ2 ) c2 r c rHS c2 r c rHS (24) Λr is the work done per unit mass in moving a test particle a distance The term GM r2 ds2 = −(1− HS Λ r by a constant force GM . This constant force is generated by the emission of a 2 rHS low energy graviton E0 = hH0 each time a Nexus graviton expands. In the weak field limit equation(24) becomes 2G(Mbar + Mvac ) 2GMΛ r 2 2 2G(Mbar + Mvac ) 2GMΛ r + 2 2 )c dt +(1+ − 2 2 )dr2 +r2 (dθ2 +sin2 θdφ2 ) c2 r c rHS c2 r c rHS (25) We implement the non-relativistic velocity requirement by taking the limit v/c → 0 This reduces equation(25) to ds2 = −(1− ds2 = −(1 − 2G(Mbar + Mvac ) 2GMΛ r 2 2 + 2 2 )c dt + dr2 + r2 (dθ2 + sin2 θdφ2 ) (26) c2 r c rHS 6 Stuart Marongwe It is evident that in nonrelativistic approximation time is curved, while space is flat.The resulting geodesic line for a test particle in this field is determined by the following differential equations µ ν d2 xλ λ dx dx + Γ =0 µν dτ 2 dτ dτ By converting equation(26) to the Cartesian form ds2 = −(1 − 2G(Mbar + Mvac ) 2GMΛ r 2 2 + 2 2 )c dt + dx2 + dy 2 + dz 2 c2 r c rHS (27) (28) we are able to conveniently calculate the Christoffel symbols Γλµν from which we obtain the final expression as dr2 G(Mbar + Mvac ) GMΛ − + 2 =0 dt2 r2 rHS (29) GMvac v2 (H0 r)2 = = = H0 v r2 r r (30) c2 (H0 rHS )2 GMΛ = = = H0 c 2 rHS rHS rHS (31) Given that and that Therefore in the weak field limit, the gravitational acceleration is GMbar d2 r = + H0 v − H0 c (32) dt2 r2 Here the last two terms refer to the acceleration contributions due to DM and DE respectively. 4.1. The Baryonic Tully-Fisher relation We now consider an aggregation of baryonic matter such as a galaxy whose baryonic mass within a radius r is M (r) and also consider a situation inwhich GM (r)bar v2 = H c = 0 r2 r We notice that under these conditions r= v2 H0 c (33) (34) and substituting for r in GM (r)bar v2 = r2 r (35) v 4 = GM (r)bar H0 c (36) we obtain The Schwarzschild Solution for the Nexus Graviton 7 which is the baryonic Tully-Fisher relation.Thus under these conditions, equation(32) reduces to dv d2 r = = H0 v dt2 dt (37) giving the final solution after integration as 1 v = V0 eH0 t = (GM (r)bar H0 c) 4 eH0 t (38) This an expression for galaxy rotation curves and evolution.The expression is also valid for late time cosmic acceleration which began once the conditions described by equation (33) were satisfied. Alternatively from equation (23), this occured when the stress momentum tensor of baryonic matter was equal to the density of DE.This state of equilibrium naturally explains the enigmatic Coincidence problem.Also from equation (38) we notice that the age of a stellar object from when condition(33) was satisfied contributes to its rotational speed.However this only becomes a significant factor if that age is comparable to Hubble time.The estimated age of the Milky Way galaxy is 13.2Gy which is comparable to the Hubble time. We therefore expect that a number of its stars will not fall neatly into the TullyFisher relation.For instance, the asymptotic speed of the Milky Way stars is about 220km/s and given that the age of the galaxy is approximately 95% of the Hubble age then we expect to find stars that have satisified condition (33) for that long, orbiting far from the galactic center at speeds of v = V0 eH0 t = 220e0.95 km/s ≈ 568km/s (39) If verified equation(38) may provide an alternative method for calculating the age of a galaxy using the rotational speed of its ancient stars. 5. Conclusion After applying the Schwarzschild approach to solving the field equations in the Nexus Paradigm of quantum gravity, we happily notice that the model is free from singularities and the ’phantom menace’.These are a nuisance in most gravitational theories and sure sign of catastrophic internal inconsistences in a theory.Thus, it is against this background the Nexus Paradigm can be seen as a viable and falsifaible paradigm of quantum gravity.Here we have a model that brings together seemingly disparate cosmic phenomena such as DM, DE,quantum gravity,late time cosmic acceleration ,the Tully-Fisher relation,galaxy rotation curves and evolution by explaining them as different manifestations of the same underlying graviton field. 6. Acknowlegements I would like to acknowlege with much appreciation the support from my family, colleagues and peers. 8 Stuart Marongwe References [1] C.Kiefer (2013) Conceptual Problems in Quantum Gravity and Quantum Cosmology Mathematical Physics Volume 2013, doi:10.1155/2013/509316 [2] C.Kiefer, M. Kramer (2012) Can effects of Quantum Gravity be Observed in the Cosmic Background? Int.J.Mod.Phys. D21, 1241001 (2012) doi:1142/S0218271812410015 [3] S. Marongwe, The Nexus Graviton: A quantum of Dark Energy and Dark Matter, International Journal of Geometric Methods in Moderm Physics Vol. 11, No. 6 (2014) 1450059 DOI: 10.1142/S0219887814500595, pp 1–20. [4] C.L.Bennett et al.(2013),Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations,. The Astrophysical Journal Supplement 208: 20. arXiv: 1212.5225, doi:10.1088/0067-0067/208/2/20 [5] C.W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, W. H. Freeman and Co. (1973), ISBN 978-0-7167-0344-0. [6] Lev D. 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