Introduction
This journal article summary is in research conducted by, Davide Batic, Marek Nowakowski and Kirk Morgan titled, “The Problem of Embedded Eigenvalues for the Dirac Equation in the Schwarzschild Black Hole Metric”. The research got published in the year 2016 in the Universe journal. Notably, the researchers sought to establish the non-existence of bound states for the Dirac equation in the Schwarzschild black hole metric. The researchers cover three major sections where the first key section is the determination and demonstration of the absence of fermion bound states around a Schwarzschild black hole. The second key section is the application of the Dirac equation in calculating the embedded eigenvalues, and the third section is the establishment of bound states regarding different mass values. This article contributes to the existing research volume by establishing flaws in oversimplified approximation in calculations using the Dirac equation in curved space-time.
Background
The researchers derived inspiration from the problem of whether bound states for the Dirac equation existed or not in the presence of a black hole. One of the loopholes pointed out in previous studies is that they relied heavily on approximations and hence availed inconclusive results. Batic et al., 2016 note that previous studies provided arguments that, “relies on an approximation of the radial system emerging from the Dirac equation after separation of variables and on the construction of approximated solutions at the event horizon and far away from the black hole”. Although there were significant manipulations on the Dirac current, the researchers emphasize that the studies experienced difficulties arising from the expression of the given conditions in the Dirac current. They also stress that since the Dirac current conditions are only be expressed in the form of elaborate sets of differential equations and inequalities, the problem on whether there exist values of particle energy as per the outline of the Dirac current (Belgiorno et al., 2009).
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On their part, the researchers assert that their aim was to apply the various differential equations and inequalities in line with the Dirac equation to determine the energy and mass of the Fermion. They also sought to integrate polynomial functions as a means of creating exponential decaying at the event horizon. They also used a “power series representation whose coefficients turn out to satisfy a four-term recurrence relation instead of a two-term recurrence retaliation” (Batic et al., 2016). Through derivation of necessary and sufficient conditions, they ensure the retention of the coefficients of the power series expansion. Eventually, the researchers use the contradiction method to prove that “there exist no real values of the energy of the Fermion satisfying the Dirac current in the Schwarzschild black hole, and, hence, no bound states can occur”( Batic et al., 2016).
Notably, Specific terms and concepts that are emphasized are eigenvalues, fermion bound states, and the Dirac equation. On looking them up through an internet search, I found that eigenvalues are transformations that act linearly on infinite-dimensional spaces (Zecca et al., 2007). More often than not, the derivation stems from infinitely differentiable real functions in differential equations. The second unfamiliar concept was fermion bound states, and I found that three types of bound states exist in fermions and result from the interaction of particles on adjacent sides of the Hubbard chain (Schmid, 2004). The first type of bound state in fermions is basis state where both fermions are on the same site, the second type is where the fermions occupy neighboring sites, and the third is where the fermions are symmetric. Therefore, the researchers used “two different exact methods to complete the proof that regardless of the value of the particle’s mass, the fermion bound states do not exist.”
Discussion
As aforementioned, the researchers used, “two different exact methods to complete the proof that regardless of the value of the particle’s mass, the fermion bound states do not exist.” In the first method, the research involved systematic manipulation of the Dirac Equation in the Schwarzschild black hole Metric. Notably, the study was repeated by the authors, that is, they were filling loopholes in already existing research documents. Some of the important studies that exist include; Finster et al. 2000, “Non-Existence of Time-Periodic Solutions of the Dirac Equation in an Axisymmetric Black Hole Geometry”. The second is Batic et al., 2008, “the bound states of the Dirac equation in the extreme Kerr metric." the third is Belgiorno et al., 2009, “Absence of Normalizable Time-periodic Solutions for the Dirac Equation in Kerr–Newman-ds black hole background.”
Given the set and type of study, that is, using differential equations, a sample population or size was not required (Belgiorno et al., 2009). The study involved a two method approach in differentiating and normalizing the Dirac equation. In the first method, the researchers engaged the manipulation of the Dirac equation in the Schwarzschild black hole metric (Batic et al., 2016). Notably, the first step involved covariant differentiation of the mass of a fermion concerning the mass of the black hole. The covariant differentiation yielded two-component spinors that gave a wave function. Similarly, the researchers associated the spinor-basis with space-time. They then normalized the resulting equation with a null tetrad as per the normalization and orthogonally relations. This succeeded operations on reducing the system differential equations that were satisfied by the radial spinors to a couple of generalized Heun equations. Subsequently, they engaged in the derivation of a set of essential and sufficient conditions for the actuality of bound state solutions for spin-1/2 particles in the Schwarzschild geometry. There was the need to repeat the equations for two times to avail completely convincing and similar findings.
The researchers applied “the deficiency index approach as their second method.” This requires researchers to construct an appropriate alteration of the radial system as a means of establishing the indices of the transformed radial operators (Zecca et al., 2007). Through the differential operator, they accounted for the number of square-integrable solutions. The succeeding operation entailed transforming the radial spinors by the initial section of the Dirac’s equation. They then integrated the condition if the radial spinors into the equation and simplified it into a second equation. Consequently, the differential operator was incorporated into the second equation yielding a third equation (Batic et al., 2006). A series of other differentiation and normalization equations followed to yield the deficiency indices for the operator. The final equation showed that “the radial system does not possess any square integrable solution on the whole real line, and, therefore, no bound states for the Dirac equation in the Schwarzschild BH metric are allowed” (Batic et al., 2016).
The variable that was held constant was the Dirac differential equation. Notably, the control variable was the symmetric Dirac current around a black hole. The main internal factor that may have impacted the outcome is the researchers’ understanding of various concepts in the Dirac equation (Schmid, 2004). The external factors that may have impacted the outcome were unverifiable since the study was theoretical. For this experiment, there were no graphs, tables, charts, and diagrams. Instead, there were differential equations. The researchers conducted various transformations on the equations using the table of spin coefficients for different choices of the null tetrad. In the first step, the Dirac equation was separated and instead, the Kinnersley tetrad was adopted. This substitution resulted in derivation of angular and radial systems as per the spin coefficients. Since the researchers were only interested in eigenvalues and associated Eigenfunctions, they separated the variables in the Dirac equation that resulted in an eigenvalue. Through the application of the eigenvalue, the equation was normalized, and eigenvalue equation obtained. The next approach was reducing the eigenvalue equation coupled with the separation of constants in the angular problem. From these equations, it becomes clear that the obtained asymptotic energy spectrum and managed to reduce a polynomial function through the radial system (Batic et al., 2016). This reduction did not possess any square integrable solution and gave way to the emergence of resonances. Therefore, the equations showed the non-existence of bound states for the Dirac equation in the Schwarzschild Black Hole metric.
Conclusion
The study gave me a deeper understanding of fermion bound states around a black hole. Since the study was focused on the existence of fermion bound states around a black hole, the conditions required zero approximations. Through meticulous differentiation and normalization of the Dirac equation in the Schwarzschild black hole metric, it was proven that claims made on the existence of bound states were inadequate. Through the application of two different exact methods, the researchers completed the proof that “regardless of the value of the particle’s mass, the fermion bound states do not exist” (Batic et al., 2016). The relevance of the results as pushed forward by the researchers are most applicable in quantum mechanics. Notably, quantum mechanics applies wave functions in specifying interaction potential of protons which exactly happens around a black hole.
References
Batic, D., Nowakowski, M., & Morgan, K. (2016). The Problem of Embedded Eigenvalues for the Dirac Equation in the Schwarzschild Black Hole Metric. Universe , 2(4), 31.
Batic, D.; Schmid, H.; Winklmeier, M. (2006). The generalized Heun equation in QFT in curved space-times. J. Phys . A, 39, 12559–12564.
Belgiorno, F.; Cacciatori, S.L. (2009). Absence of Normalizable Time-periodic Solutions for the Dirac Equation in Kerr–Newman-black hole background. J. Phys . A, 42, 135207.
Schmid, H. (2004). Bound State Solutions of the Dirac Equation in the extreme Kerr Geometry. Math. Nachr. 274–275, 117–129.
Zecca, A. (2007). Spin 1/2 bound states in Schwarzschild geometry. Adv. Studies Theor. Phys, 1, 271–279. 2.
