what was einstein reaction to the idea of schwarzschild about spacetime around a spherical star
Twisted spacetime in Einstein gravity
Abstract
Nosotros find a vacuum stationary twisted solution in four-dimensional Einstein gravity. Its frame dragging angular velocities are antisymmetric with respect to the equatorial plane. Information technology possesses a symmetry of joint inversion of time and parity with respect to the equatorial airplane. Its Arnowitt-Deser-Misner (ADM) mass and angular momentum are nil. It is curved but regular all over the manifold. Its Komar mass and Komar athwart momentum are also nix. Its infinite red-shift surface coincides with its issue horizon, since the event horizon does non rotate. Furthermore we extend this solution to the massive case, and observe some similar backdrop. This solution is a stationary axisymmetric solution, simply non Kerr. It explicitly proves that pure Einstein gravity permits dissimilar rotational mode other than Kerr. Our results demonstrate that the Einstein theory may have much more rich structures than what we ever imagine.
blackness hole; Einstein gravity
I Introduction
Einstein'due south general relativity ein is the standard modern gravity theory. Exact solutions take pivotal condition in the theory. The well known three classical tests of general relativity depend on the Schwarzschildsolution schw , which play decisive roles in the early on developments of the full general relativity. Many verbal solutions for Einstein equation have been found exa . Among these exact solutions, two of the most significant solutions are Schwarzschild and Kerr kerr . The Schwarzschild solution describes the spacetime around a spherically symmetric star. The Kerr solution describes the spacetime around a rotating star. Actually, the Schwarzschild and Kerr solutions can describe the gravitational fields in vast ranges from millimetre to the clusters of galaxies (the Newton's law as a weak field limit).
Information technology is generally believed that the Schwarzschild solution is the unique static vacuum solution, and the Kerr solution is the unique rotating vacuum solution on an asymptotic manifold in Einstein gravity afterward the proof of the uniqueness theorems, for reviews encounter uni . Nigh all the celestial objects are rotating, quickly or slowly. So the Kerr solution is extremely of import in the studies of astrophysics. Unfortunately we cannot obtain a completely satisfactory interior solution matching to Kerr upward to at present, though great efforts accept been evolved in this topic for more than than l years. On the opposite, we obtain several negative results. For examples, a perfect fluid cannot be the source of Kerr her , and the analytic estimate solution that describes the slowly rotating astrophysical object does not lead to Kerr bos . The other problem of the Kerr solution is that information technology is unstable against linear perturbations in the interior region teu . That ways the result of collapsing of a rotating star may not be a Kerr black hole, even if the outside region of the progenitor can be described past Kerr solution. So, theoretically the question is: Are there any rotating modes unlike from Kerr, more or less similar the wave functions of the hydrogen cantlet with the aforementioned athwart momentum?
At the get-go sight the answer is no, since the Carter uniqueness theorem forbids them machine . Exactly, the Carter uniqueness theorem requires: a. the manifold is axially symmetric and stationary; b. the manifold is asymptotically apartment, and the full mass and full angular momentum measured at the infinity, i.e., the ADM mass and angular momentum adm , are respectively; c. the manifold is regular everywhere at the exterior region of the horizon (including the horizon). Under these weather condition, the spacetime must exist Kerr. This is a quite exciting but harsh theorem. It almost determines the metric effectually any celestial object (near all celestial bodies are rotating), including planet, stars, galaxies, and cluster of galaxies. It is just Kerr. Yet, as we have mentioned, Kerr has some issues. Interestingly, by a careful analysis of the Israel uniqueness theorem isr and the Carter uniqueness theorem, nosotros find that the Einstein equation permits more vacuum asymptotically flat rotating solution other than Kerr.
We explicate the general thought for our approach on a rotating spacetime to evade the uniqueness theorems. We consider a rotating spacetime with . Since information technology is rotating the Israel uniqueness theorem say nada most this instance. At the same time its full angular momentum is zero, thus the Carter uniqueness neither requires it to be Kerr. Our fundamental idea is that "rotating" does not contradict with , which is beyond the traditional lore. For instance, we can consider the example that the space rotates at different directions in unlike spherical shells. The angular momentums in dissimilar shells are bundled to be exactly counteracted, and thus the total angular momentum vanishes. Without breaking the centric symmetry, we require the directions of rotation of the shells to be up or down. We observe a twisted solution which is a little more complicated than this heuristic example.
The metric read,
where is a constant, and ( ) are spherical coordinates, which come back to the standard spherical coordinates in Minkowski spacetime when . The Ricci tensor for this metric vanishes,
Thus it is a vacuum solution of the Einstein equation. It is a curved infinite and regular all over the manifold, since the only not-zero scalar polynomial of curvatures (the Kretschmann scalar) reads,
Information technology is like shooting fish in a barrel to check that the spacetime (1) possesses a discrete symmetry of articulation transformations of fourth dimension inversion ( ) and reflection with respect to the equatorial aeroplane ( ).
The infinite red-shift surface dwells at
And the event horizon constant satisfies,
It is easy to obtain that the infinite red-shift surface coincides with the event horizon at . We shall discuss the the reason for this coincidence later. It is a blackness hole, since information technology has an event horizon. Yet, it is very unlike from the ordinary holes, which accept spacetime singularities. From (3) 1 sees that the black hole (one) is regular everywhere in the whole spacetime.
The frame dragging angular velocity reads,
Based on the detailed discussions on the rotating metrics, 1 finds that the frame dragging velocity can be treated as the athwart velocity of the spacetime itself rot . A simple caption is that a co-rotating observer,
will sense a time-orthogonal spacetime like a static one (without space-time cross term). At the horizon , the frame dragging velocity vanishes. In this sense, the horizon is static. Thus the space red-shift surface coincides with the result horizon. That is different from the case of Kerr, for which the outcome horizon is rotating, and then that the infinite crimson-shift surface is separated from the event horizon. The other important belongings of the frame dragging angular velocity is that it vanishes at the equatorial aeroplane , and is antisymmetric with respect to the equatorial plane, i.e., .
Fixing , for the large approximation we have,
Hence, has at least one maximum for . In this interval only 1 signal satisfies,
At this point,
From the previous discussions, we obtain the picture of this spacetime. It is a twisted spacetime. The spacetime rotates in opposite directions in a higher place and below the equatorial airplane. The equatorial plane itself does not rotate. A sketch of this spacetime is shown in the fig. ane. If 1 inserts an elastic bar with finite thickness along , it will be twisted to be something like a spiral steel bar.
Next we written report the mass and athwart momentum of this spacetime. Get-go we prove that the spacetime (1) is asymptotically flat. For the sake of canceling the frame dragging effects, we introduce the co-rotating coordinates,
With this new coordinate system, the metric (1) reads,
where correspond the exponents of metric (1). Hither they are treated as the functions of ( ). One tin evidence that ( ) are (quasi-)spherical coordinates in the ADM formalism, which satisfies
where are the components of Minkowski metric in spherical coordinates. According to the standard formulae of ADM mass and ADM angular momentum, we obtain
and
The ADM formulism can only obtain the total mass and athwart momentum. It cannot say annihilation about the spatial distributions of the mass and angular momentum. The distribution of mass and angular momentum of gravity field is a very intricate trouble. A local mass seems necessary in the cases, for example, the propagation of gravitational waves from the source to the local observer. However, an energy-stress density is prohibited past the equivalence principle. Thus we turn to the quasi-local forms every bit an inevitable concession. Later on decades' studies we have several dissimilar definitions quasi-local gravitational mass and angular momentum. Commonly, they are not equal to each other in the same 2-surface. Some of them are meaning non only in the studies of mass of gravity, but likewise in other topics including thermodynamics, exact solutions etc self . Hither we do not check them 1 by one for the metric (1). As an example we study one of the earliest form, the Komar integral, which is applicative in the stationary spacetime kom . The Komar integral defines the gravitational mass by the imprints of gravitational furnishings on a 2-surface in a stationary spacetime. Originally, it is presented as the total mass for a space-like world canvass in an infinite 2-surface. In principle, information technology is too tin can be used to define a quasi-local mass in a finite 2-surface,
where a star denotes the Hodge dual operator, and is the lower index form of the time similar Killing vector,
For the metric (1), the Komar integral (16) in 2-surface of radius presents,
The Komar mass vanishes when . This result is consequent with the ADM mass (14). The concrete interpretation is that the kinetic free energy (because of rotation) exactly counteracts the potential energy. For a finite ii-surface, the Komar mass always larger than aught. This ways that the kinetic energy always larger than the potential energy. The Komar angular momentum reads,
where
In a two-surface of radius , the Komar angular momentum reads,
This result is also consequent with the ADM angular momentum. Further, information technology presents a more fine issue, which is contained of . The Physical interpretation is that the angular momentum above the equatorial airplane exactly counteracts the athwart momentum below the airplane. Of course, information technology embodies the property of antisymmetry of the frame-dragging athwart velocity with respect to the equatorial airplane. So, we call this solution rotating space without (total) mass and (full) angular momentum. There is but one parameter in this solution, which denotes the angular velocity of the spacetime. In the following text, we shall introduce 1 more than parameter into the metric (1), which tin exist interpreted as the total mass of a spacelike world sheet.
And so nosotros brand a preliminary discussion about the motion of a exam particle in the spacetime (ane). First we consider a particle moving on the equator plane . For a fourth dimension-similar geodesics, we obtain at large approximation,
Here , and is the athwart momentum of the test particle. Note that is different from the angular momentum of the spacetime . Comparing to the equation of motion in the Schwarzschild spacetime, we find that the effective Schwarzschild mass is , and the leading correction term becomes rather than .
For more realistic studies, we extend the solution (1) to a solution with one more than parameter ,
The metric (23) is a solution of vacuum Einstein equation. For it the Ricci tensor reads,
It is easy to come across that the metric (23) comes dorsum to metric (1) when . Also one can check that (23) becomes Schwarzschild metric when . The space red-shift surface and consequence horizon coincide at,
The metric (23) is however regular all over the spacetime, since the Kretschmann scalar reads,
The properties of the frame-dragging velocity are also like to the massless example. In the dragged frame (eleven), one can demonstrate the asymptotic flatness of the spacetime (23). And so the ADM mass and athwart momentum read,
and
respectively. The Komar mass and angular momentum in a 2-surface of radius read,
and
respectively. Nosotros meet that is the mass parameter, which reduces to the Schwarzschildmass when , and the full angular momentum is goose egg. The physical interpretations of these results follow the previous case. Mimicking the previous massless case, we obtain the equation of geodesics for a test particle moving on the equatorial plane at large approximation,
In this form, we recover all the terms for the Schwarzschild solution (the terms at the left hand side of the above equation). The correction terms relative to the Schwarzschild spacetime are casted to the right paw side of the (31).
In summary, we obtain a vacuum stationary asymptotically flat axisymmetric twisted spacetime in Einstein gravity and study some preliminary properties of it. It describes a rotating spacetime with total angular momentum . The directions of the angular velocities of the spacetime are antisymmetric with respect to the equatorial plane. So its concrete image is a twisted spacetime. There is no true singularity in this spacetime. Its ADM mass and angular momentum, and full Komar mass and athwart momentum are nil. We study the motion of a test particle on the equatorial plane and find the equation of motion. Finally, we present a massive version of this twisted solution. Commonly, the Carter uniqueness theorem of black pigsty is alleged to be "a stationary asymptotically apartment axisymmetric solution of Einstein gravity must be Kerr." In this sense, we find a counterexample of this merits. Through detailed analysis of the theorem nosotros notice that our upshot in fact does not violate the Israel and Carter theorems. To observe the observational indications is the future work. This Letter implicitly displays that the Einstein gravity may notwithstanding hide more amazing structures beyond our present studies.
Ii Acknowledgements
This piece of work is supported by the Programme for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, National Didactics Foundation of People's republic of china under grant No. 200931271104, and National Natural Science Foundation of China under Grant No. 11075106 and 11275128.
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Source: https://www.arxiv-vanity.com/papers/1609.09721v1/
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