Exploring the Nature of Black Hole Singularities: An Investigation into the Fate of an Infalling Observer
The study aims to understand the effects of these regions of extreme gravitational strength, i.e. singularities on an extended body following a time-like geodesic that encounters a singularity. The study answers the following questions:
- What is the fate of an extended body as it moves along a time-like geodesic that falls into a Schwarzchild blackhole?
- What is the nature of the singularity that lies at the centre of the Schwarzchild black hole?
- Can Einstein’s equations yield a generic class of solutions that admit a null and weak space-time singularity?
- What is the fate of an extended body as it moves along a time-like geodesic and falls into a null and weak space-time singularity?
General relativity has been a well established theory for more than a hundred years, it provided a theoretical framework that revolutionised our knowledge and understanding of the Universe to great depths. The basis of science lies in the recognition of flaws in a theory and work towards a more complete theory. Einstein recognized the flaws in Newton’s theory of gravity and this motivated him to formulate his own theory of gravity, which has proved to be remarkable. Similarly, It is well known that Einstein’s equation breaks down and yields ambiguous results in extreme conditions of gravity, signalling the limitations of this theory. A region of such interest is a singularity that lies within black holes. It has been long known in theory that objects such as black holes can exist but it is now known through observational evidence that black holes do exist. The Event Horizon Telescope was the first to capture a photograph of a black hole in 2017, in this case the one residing at the heart of M87 Galaxy. Further, singularity theorems imply that a singularity is expected to develop inside a black hole. Even after such strong evidence of the existence of spacetime singularities, there is very little known about them. Hence, the study of singularities provides us with an opportunity to uncover a new layer of reality that emerges under extreme conditions of gravity.
The study starts with getting to grips with the calculation of tensor’s associated with general relativity such as the Riemann curvature tensor and the Ricci tensor. Further, the geodesic equations and the geodesic deviation equations for a radially infalling time-like geodesic are formulated to understand the behaviour of the Jacobi fields as a body falls into a Schwarzchild singularity. The geodesic deviation equations provide us with a set of second order differential equations that describe the behaviour of Jacobi fields. We interpret these Jacobi fields to form a 3-dimensional body and assess the behaviour of the individual Jacobi fields by solving the second order differential equations which shows how the dimensions of the body change as it moves through the geodesic falling into the singularity. Moving ahead, through rigorous mathematical calculations using Wolfram Mathematica this study proves the mathematical framework that describes the existence of a generic class of solutions to the Einstein field equations that admit the presence of a null space-time singularity. The geodesic and geodesic deviation equations are formulated in a 2-dimensional space with the same constraints that were used to prove the existence of a weak space-time singularity.
The definition of our parameter s is such that, at s=0, the geodesic congruence encounters a singularity. We can see that as s approaches zero, V(s) approaches zero. Hence, as we move along the congruence, the volume encompassed by the Jacobi fields tends to zero. For an observer falling into a black hole, we can conclude that an observer would be crushed completely. Hence, calculating the volume of these Jacobi fields at s=0 provides us a way to examine the nature of the singularity. In the case of a Schwarzchild black hole, The volume goes to zero at the point of singularity; the nature of the singularity is deemed to be strong and space-like. Further, we can look at the individual behaviour of the Jacobi fields at s=0 as well.
We can infer that an observer falling into a Schwarzschild black hole will be infinitely stretched along the radial direction and will be completely crushed along the tangential and azimuthal directions. This phenomenon is called Spaghettification, where the tidal forces create a gradient of gravitational attraction across your body. The side of the body closer to the black hole experiences a stronger gravitational pull than the other side. This difference in gravitational pull can stretch and compress the body.
Additionally, The study through meticulous mathematical analysis fills in the claims used by Ori and Flanagan to established the presence of a generic set of solutions to Einstein’s Field Equations. These solutions encompass a particular class that displays a weak and null singularity at the v=0 hypersurface. Furthermore, we’ve developed the geodesic equations and geodesic deviation equations within a two-dimensional spatial context. These equations pertain to the motion of a timelike geodesic as it descends towards the v=0 hypersurface.
A cornerstone of this study lies in the insights drawn from prior research in the realm of black hole physics and spacetime singularities. The investigation starts with comprehending the nature of the singularity residing at the heart of a Schwarzschild black hole, serving as a fundamental example. This exploration then extends its reach into the focal point of this research, a more intricate and rigorous realm of functionally generic null and weak singularities. In this part, we proved the presence of the corresponding singularity and derived the geodesic and geodesic deviation equations, showcasing how the geodesic congruences are affected in regions of singularities. The heart of this study involves meticulous mathematical analysis, drawing insights from diverse mathematical disciplines such as tensor calculus, differential geometry, and partial differential equations. These varied tools of mathematics are then practically employed through the utilization of Mathematica, a user-friendly software by Wolfram that provides an accessible means of executing computational mathematics. The objective of this research was to fill in the mathematical propositions presented in [1], a work that has remained unpublished. This study achieved its goal by crafting a clear and concise formulation for these mathematical assertions. Through these concerted efforts, this study has bridged the gap in knowledge, enriching the discourse in this domain.
As with any study, the path undertaken in this research is not devoid of limitations. The limitations encompassed in this work—whether stemming from the scope of mathematical formalism, the intricacies of theoretical assumptions or time—serve as signposts for future explorations. These limitations, while guiding the trajectory of this study, also pave the way for future scholars to tread further upon this trajectory, to unveil even greater layers of insight and understanding. An essential aspect to acknowledge in our research lies in the constraint of our construction to analytic initial functions. Although it is recognized that this is a technical limitation arising from the mathematical theorems. However, I firmly believe that the physical scenario we’ve explored—a null weak singularity—will indeed manifest even if the initial functions on S are not analytic. This belief holds, provided that these initial functions remain adequately smooth for v < 0 [1]. It’s noteworthy that the progression of time has also exerted an influence on the trajectory of our research, influencing the depth to which we could delve. As we conclude this phase of investigation, we recognize the potential for future explorations to transcend these limitations, unveiling even greater depths of understanding in the captivating realm of theoretical physics.
[1] Ori, A., & Flanagan, E. E. (1996). How generic are null spacetime singularities?. Physical Review D, 53(4), R1754.
[2] Tipler, F. J. (1977). Singularities in conformally flat spacetimes. Physics Letters A, 64(1), 8-10.
[3] Poisson, E., & Israel, W. (1990). Phys. Rev. D 41, 1796.
[4] Ori, A. (1991). Phys. Rev. Lett. 67, 789.
[5] Bonano, A., et al. (1995). Proc. R. Soc. London A450, 553; Brady, P. R., & Smith, J. D. (1995). Phys. Rev. Lett. 75, 1256.
[6] Ori, A. (1992). Phys. Rev. Lett. 68, 2117
[7] Yurtsever, U. (1993). Class. Quantum Grav. 10, L17.
[8]. Flanagan, B. E., & Ori, A. (unpublished).
[9] Belinsky, V. A., & Khalatnikov, I. M. (1970). Sov. Phys. JETP 30, 1174; Khalatnikov, I. M., & Lifshitz, E. M. (1970). Phys. Rev. Lett. 24, 76; Belinsky, V. A., Khalatnikov, I. M., & Lifshitz, B. M. (1970). Adv. Phys. 19, 525.
[10] Poisson, E. (2004). A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press. Figure 1.1, Chapter 1, Page 3.
[11] Poisson, E. (2004). A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press. Figure 1.3, Chapter 1, Page 17.
[12]Poisson, E. (2004). A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press. Figure 2.5, Chapter 2, Page 41.