aa r X i v : . [ phy s i c s . g e n - ph ] M a y Physical Metric and the Nature of Gravity
Yukio Tomozawa ∗ Michigan Center for Theoretical Physics and Randall Laboratory of Physics,University of Michigan, Ann Arbor, MI. 48109-1120, USA (Dated: September 24, 2018)
Abstract
A physical metric is defined as one which gives a measurable speed of light throughout the wholespace time continuum. It will be shown that a metric which satisfies the condition that speed oflight on the spherical direction is that in a vacuum gives a correct result. All the metric functionsthus obtained are positive definite and exhibits a repulsive force at short distances. The horizonin the sense of vanishing of the speed of light still exists in the radial direction. It is located at3 √ r s / . r s , where r s = 2 GM/c is the Schwarzschild radius. This radius corresponds tothe size of a black hole, as well as the photon sphere radius. The metric can be used to calculategeneral relativistic predictions in higher order for any process. ∗ Electronic address: [email protected] . INTRODUCTION The Schwartzschild metric is the exact solution of the Einstein equation of generalrelativity[1],[2]. It is a metric for the spherically symmetric and static (SSS) system. How-ever, since the speed of light in a spherical direction inside the horizon of the Schwarzschildmetric is imaginary, it is not a physical metric. This characteristic, of course, is changed ifthe role of r and t is awitched inside the horizon. The speed of light becomes positive defi-nite even inside the horizon. Then, the meaning of static is lost, i. e. the metric inside thehorizon is non-static. In this article, the author constructs a physical metric by a coordinatetransformation of the Schwarzschild metric, maintaining the characteristic of static natureand discusses the nature of gravity in the obtained physical metric. In a forthcoming article,the author discusses the difference between the Schwarzschild metric and the physical metricfor the prediction of time delay experiment of Shapiro et.al[3].
II. ASYMPTOTIC FORM FOR THE PHYSICAL METRIC
The physical metric is expressed as ds = e ν ( r ) dt − e λ ( r ) dr − e µ ( r ) r ( dθ + sin θdφ ) , (1)for a mass point M in a spherically symmetric and static (hereafter refered as SSS) metric.From the fact that the transformation, r ′ = re µ ( r ) / , leads to the Schwarzschild metric[4],one can deduce the expression for the metric, e ν ( r ) = 1 − ( r s /r ) e − µ ( r ) / , (2) e λ ( r ) = ( ddr ( re µ ( r ) / )) / (1 − ( r s /r ) e − µ ( r ) / ) , (3)where r s = 2 GM/c is the Schwarzschild radius. An asymptotic expansion for the metricfunctions can be obtained from Eq.(2) and Eq.(3), yielding e ν ( r ) = ∞ X n =0 a n ( r s /r ) n , e λ ( r ) = ∞ X n =0 b n ( r s /r ) n , and e µ ( r ) = ∞ X n =0 c n ( r s /r ) n , (4)where a = b = c = 1 , (5) − a = b = 1 and (6) a = c / , b = 1 − c / c / − c , etc. (7)2t is obvious that a n +1 and b n can be expressed as functions of c n , c n − . . . , c . III. CONSTRUCTION OF THE PHYSICAL METRIC IN THE ASYMMPTOTICREGION
Since the trouble of the Schwarzschild metric lies in the speed of light on a sphericaldirection inside the horizon, one can eliminate this trouble by requiring the following ansatz.
Proposition 1
The speed of light in the angular direction in the SSS metric is the same asthat of vacuum.
In other words, we require e ν ( r ) = e µ ( r ) = ω (8)This ansatz implies that although gravity deforms the geometry of space-time, speed of lightperpendicular to the gravity will not be affected.Then one gets the equation for the asymptotic solution, e ν ( r ) = 1 − ( r s /r ) e − µ ( r ) / = e µ ( r ) = ω. (9)Then one has r s /r = e µ ( r ) / (1 − e µ ( r ) ) = ω / (1 − ω ) , (10)or ( r s /r ) = ω (1 − ω ) . (11)Differentiating Eq.(11), one gets r dωdr = 2 r s /r (1 − ω )(3 ω −
1) = 2 ω (1 − ω )(3 ω − . (12)From Eq.(3), the metric function in the radial direction can be calculated e λ ( r ) = ( ddr ( rω / )) /ω = ( ω / + ω − / r dωdr / /ω = ( 2 ω ω − . (13)From Eq.(10) or Eq.(11) and Fig. 1, it is clear that one covers the range of1 > ω > / ∞ > r/r s > √ / . (15)In order to cover the range of r/r s < √ / , (16)one has to use non-asymptotic solution of the Schwarzschild solution. From Appendix, sucha solution is given in the next section. 4he asymptotic expansion of the metric functions can be calculated from Eq.(11) andEq.(13) as ω = e ν ( r ) = e µ ( r ) = 1 − ( r s /r ) −
12 ( r s /r ) −
58 ( r s /r ) − ( r s /r ) − ..... (17)and e λ ( r ) = 1 + ( r s /r ) + 94 ( r s /r ) + 438 ( r s /r ) + 21116 ( r s /r ) + ...... (18)where Eq.(13) has been used. Successive expansion yields a determination of all the param-eters, c n , for the physical metric. These are useful for testing obervational data in higherorder in gravity. Alternatively, the inverse function of Eq.(10) or Eq.(11) may be used. IV. THE PHYSICAL METRIC IN THE WHOLE REGION
The Schwarzschild solution for non-asymptotic region (See Appendix section) can bewritten as e λ ( r ) = (1 + Dr s r ′ ) =1 (19)and e ν ( r ) = 1 A (1 + Dr s r ′ ) , (20)where A and D are non-dimensional constants. The metric functions for the physical metricin the region r/r s < √ / ω = e ν ( r ) = 1 A (1 + D ( r s /r ) e − µ ( r ) / ) = e µ ( r ) (22)and e λ ( r ) = ( ddr ( rω / )) /Aω, (23)and hence D ( r s r ) = ω / ( Aω −
1) (24)or ( Dr s r ) = ω ( Aω − (25)Differentiating Eq.(24), one gets r dωdr = − ω Aω − Aω − e λ ( r ) = ( ω / + r dωdr / ω / ) /Aω = A ( 2 ω Aω − (27)Imposing the continuity of the asymptotic expression, Eq.(10) and the non-asymptoticexpression, Eq.(24) at ( r/r s , ω ) = (3 √ / , /
3) (28)one gets A = 2 D + 3 . (29)One splits the parameter space in the following 3 Regions.I) A >
D > , II) 3 > A > > D > − III) 0 > A and − > D .In Region I, the distance r can be reached at zero when ω reaches ∞ , as ω = ( Dr s Ar ) / (30)In Region II, the distance r cannot reach at zero value (the origin). In Region III, thedistance r can be reached at zero, as in Eq.(30). However, the radial metric function, e λ ( r ) ,becomes negative as is seen from Eq.(23) or Eq.(27). This means that the radial velocityof light becomes imaginary. Therefore the most natural choice for the parameter space isthe Region I, where all metric functions are positive definite. This is a characteristic of thephysical metric. V. THE NATURE OF THE PHYSICAL METRIC
Summarizing the previous sections, one obtains the physical metric e ν ( r ) = e µ ( r ) = ω (31)in the following manner. For the asymptotic region r/r s ≥ √ / , (32)one has r s /r = ω / (1 − ω ) (33)6nd e λ ( r ) = ( 2 ω ω − . (34)The range of ω is restricted to ω ≥ / . (35)For the non-asymptotic region 3 √ / ≥ r/r s > , (36)one has D ( r s r ) = ω / ( Aω −
1) (37)and e λ ( r ) = A ( 2 ω Aω − . (38)The continuity of Eq.(33) and Eq.(37) at( r s /r, ω ) = (3 √ / , /
3) (39)requires A = D/ . (40)Choosing the range of the parameter space to be D > , and A > , (41)all the metric functions are positive definite, which guarantees the condition for the phys-ical metric, having the definite value for speed of light throughout the whole space-timecontinuum.At the origin, the metric function, ω = g ( r ), diverges as ω = ( Dr s Ar ) / . (42)From Eq.(34), one concludes that e λ ( r ) = g ( r ) becomes ∞ at the edge point of the asymp-totic region, Eq.(39). Hence, at r s /r = 3 √ /
2, the radial speed of light vanishes. One maycall this point a horizen, although the characteristics are very different from that of theSchwartzscield metric. The speed of light in the spherical direction is that of vacuum. Sinceall metric are positive definite in the whole region, speed of light is well defined throughoutthe whole space time. The magnitude of the horizon is (3 √ / r s = 2 . r s , i.e., 2.6 times7igger than that of the Schwartzschild radius. Below Fig. 2 showes the picture of g ( r ) = e ν ( r ) = ω as a function of r/r s , namely the picture of the gravitational potential with theshift of the y axis and a scale factor of 2.The crossing point of the two curves is( r/r s , ω ) = (3 √ / , / . (43)The radial metric function, g = e λ ( r ) , is inevitably discontinuous at this point. Thisdiscontinuity allows the passing of all particles through the horizon, in and out. More8mportantly, the gravitational potential inside the horizon is repulsive. This property couldchange the nature of gravity, black holes, cosmic ray production as well as the nature ofcosmology. I will discuss these problems in the forthcoming articles. VI. SUMMARY
The author has constructed a physical metric for which the speed of light is well definedthroughout all space time continuum. The constructed metric shows that gravity is repulsiveinside the horizon and the size of black holes is 3 times bigger than the Schwarzschild radius.The latter can be tested by observation in the near future[5]. The observational effects ofthe physical metric will be discussed in forthcoming articles.
VII. APPENDIX. THE SCHWARZSCHILD SOLUTION
Setting e µ ( r ) = 1 , (44)in Eq.(1), and using the Maple program the Einstein equation reads − rλ ′ ( r ) − e λ ( r ) + 1 = 0 , (45) − rν ′ ( r ) + e λ ( r ) − ν ′ ( r ) − λ ′ ( r ) + 2 rν ′′ ( r ) + rν ′ ( r ) − rν ′ ( r ) λ ′ ( r ) = 0 . (47)From the sum of Eq.(45) and Eq.(46), one gets ν ′ ( r ) + λ ′ ( r ) = 0 . (48)Using this relation, Eq.(47) becomes − rλ ′′ ( r ) + rλ ′ ( r ) − λ ′ ( r ) = 0 (49)or equivalently e λ ( r ) ( re − λ ( r ) ) ′′ = 0 . (50)9n the other hand, Eq.(45) can be written as( re − λ ( r ) ) ′ = 1 , (51)which solution is e − λ ( r ) = 1 + Br , (52)and Eq.(50) is satisfied, where B is an integration constant. The solution of Eq.(48) reads e ν ( r ) = 1 A (1 + Br ) . (53)The asymptotic solution with the boundary condition is given by A = 1 , B = − r s . (54)On the other hand, the non-asymptotic solution is given by B = Dr s , A arbiray. (55)where A and D are nondimensional integration constants. Acknowledgments
It is a great pleasure to thank Peter K. Tomozawa for reading the manuscript. [1] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, (Freeman, San Francisco, 1973)[2] S. Weinberg, Gravitation and Cosmology, (Wiley and Sons, New York, 1972), Eq. (8.7.4)[3] I. I. Shapiro et. al., Phy. Rev. Letters 17, 933 (1966); R. D. Resenberg and I. I. Shapiro, Apj.234, L219 (1979); B. Bertotti, L.Iess and P. Tortora, Nature 425, 374 (2003)[4] K. Schwarzschild, Sitzungsber. Press. Skad. Wiss. Berlin Math. Phys.) 189-196 (1916)[5] S. S. Doeleman et. al., Science 338, 335 (2012)
Figure captionsFig. 1 The metric function, g ( r ), in the asymptotic region in the SSS physical metric.Fig. 2 The metric function, g ( r ), as a funtion of r/r ss