When one presumes that the gravitational mass of a neutral massenpunkt is finite, the Schwarzschild coordinates appear to fail to describe the region within the event horizon (EH), of a Schwarzschild Black Hole (SBH)....When one presumes that the gravitational mass of a neutral massenpunkt is finite, the Schwarzschild coordinates appear to fail to describe the region within the event horizon (EH), of a Schwarzschild Black Hole (SBH). Accordingly, the Kruskal coordinates were invented to map the entire spacetime associated with the SBH. But it turns out that at the EH (Mitra, IJAA, 2012), and the radial timelike geodesic of a point particle would become null. Physically this would mean that, the EH is the true singularity, i.e., M = 0, and this zero mass BH could only be a limiting static solution which must never be exactly realized. However, since in certain cases , here we evaluate this derivative in such cases, and find that, for self-consistency, one again must have at the EH. This entire result gets clarified by noting that the integration constant appearing in the vacuum Schwarzschild solution (and not for a finite object like the Sun or a planet), is zero (Mitra, J. Math. Phys., 2009). Thus though the Schwarzschild solution for a point mass is formally correct even for a massenpunkt, such a point mass or a BH cannot be formed by physical gravitational collapse. Instead, physical gravitational collapse may result in finite hot quasistatic objects asymptotically approaching this ideal mathematical limit (Mitra & Glendenning, MNRAS Lett. 2010). Indeed “the discussion of physical behavior of black holes, classical or quantum, is only of academic interest” (Narlikar & Padmanbhan, Found. Phys. 1989).展开更多
The total spacetime manifold for a Schwarzschild black hole (BH) is believed to be described by the Kruskal coordi-nates and , where r and t are the conventional Schwarzschild radial and time coordinates re-spectively...The total spacetime manifold for a Schwarzschild black hole (BH) is believed to be described by the Kruskal coordi-nates and , where r and t are the conventional Schwarzschild radial and time coordinates re-spectively. The relationship between r and t for a test particle moving along a radial or non-radial geodesic is well known. Similarly, the expression for the vacuum Schwarzschild derivative for a geodesic, in terms of the constants of motion, is well known. However, the same is not true for the Kruskal coordinates;and, we derive here the expression for the Kruskal derivative for a radial geodesic in terms of the constants of motion. In particular, it is seen that the value of ) is regular on the Event Horizon of the Black Hole. The regular nature of the Kruskal derivative is in sharp contrast with the Schwarzschild derivative, , at the Event Horizon. We also explicitly obtain the value of the Kruskal coordinates on the Event Horizon as a function of the constant of motion for a test particle on a radial geodesic. The physical implications of this result will be discussed elsewhere.展开更多
文摘When one presumes that the gravitational mass of a neutral massenpunkt is finite, the Schwarzschild coordinates appear to fail to describe the region within the event horizon (EH), of a Schwarzschild Black Hole (SBH). Accordingly, the Kruskal coordinates were invented to map the entire spacetime associated with the SBH. But it turns out that at the EH (Mitra, IJAA, 2012), and the radial timelike geodesic of a point particle would become null. Physically this would mean that, the EH is the true singularity, i.e., M = 0, and this zero mass BH could only be a limiting static solution which must never be exactly realized. However, since in certain cases , here we evaluate this derivative in such cases, and find that, for self-consistency, one again must have at the EH. This entire result gets clarified by noting that the integration constant appearing in the vacuum Schwarzschild solution (and not for a finite object like the Sun or a planet), is zero (Mitra, J. Math. Phys., 2009). Thus though the Schwarzschild solution for a point mass is formally correct even for a massenpunkt, such a point mass or a BH cannot be formed by physical gravitational collapse. Instead, physical gravitational collapse may result in finite hot quasistatic objects asymptotically approaching this ideal mathematical limit (Mitra & Glendenning, MNRAS Lett. 2010). Indeed “the discussion of physical behavior of black holes, classical or quantum, is only of academic interest” (Narlikar & Padmanbhan, Found. Phys. 1989).
文摘The total spacetime manifold for a Schwarzschild black hole (BH) is believed to be described by the Kruskal coordi-nates and , where r and t are the conventional Schwarzschild radial and time coordinates re-spectively. The relationship between r and t for a test particle moving along a radial or non-radial geodesic is well known. Similarly, the expression for the vacuum Schwarzschild derivative for a geodesic, in terms of the constants of motion, is well known. However, the same is not true for the Kruskal coordinates;and, we derive here the expression for the Kruskal derivative for a radial geodesic in terms of the constants of motion. In particular, it is seen that the value of ) is regular on the Event Horizon of the Black Hole. The regular nature of the Kruskal derivative is in sharp contrast with the Schwarzschild derivative, , at the Event Horizon. We also explicitly obtain the value of the Kruskal coordinates on the Event Horizon as a function of the constant of motion for a test particle on a radial geodesic. The physical implications of this result will be discussed elsewhere.
