Papers by Carlos Castro Perelman
After reviewing the basics of Non-inertial relativity theory based on the existence of a maximal ... more After reviewing the basics of Non-inertial relativity theory based on the existence of a maximal proper force b, it allowed to postulate a modified Newtonian attractive gravitational force (and potential) which is f inite at the origin : |F (r = 0)| = b, and which vanishes at r = ∞. Secondly, from the modified gravitational potential energy we were able to glean the expression for a running gravitational coupling G(r) which exhibits asymptotic-freedom-like properties : G(r = 0) = 0, and G(r = ∞) = G_N. No quantum corrections were necessary to decrease the strength of gravity at short distances. Thirdly, we found that for very large masses m_1, m_2 (compared to √ b) the threshold in the values of r obeying κr^2 << 1, where the non-Newtonian regime becomes manifest, becomes larger and larger as m_1, m_2 become larger and larger. Whereas for very small masses (compared to √ b) the threshold in the values of r obeying κr^2 << 1, where the non-Newtonian regime becomes manifest, becomes smaller and smaller as m_1, m_2 become smaller and smaller. In the b = ∞ limit one recovers the Newtonian gravitational force for all values of r > 0. These results were all possible by abandoning the weak equivalence principle at short distances.
We pursue further our work on (Born Reciprocal) Non-inertial Relativity theory. Starting with a b... more We pursue further our work on (Born Reciprocal) Non-inertial Relativity theory. Starting with a brief review of the theory, and how Noninertial Relativity redefines the notion of mass, the phase space particle trajectories in D = 2 + 2 are revisited by emphasizing the key difference between a truly U (1, 1)-invariant mass M and the Lorentz-invariant mass m. The derivation of the generalization of the special relativistic expression of E = m(1-v 2)^{-1/2} (c = 1) to the non-inertial relativistic case follows. This, in turn, leads to the non-inertial relativistic extension of Milgrom's modified Newtonian dynamics (MOND) law. In the most general setting, one finds proper-time m(τ), and spacetime-dependent m(x^µ) masses for point particles, when the proper force F depends on τ or x^µ , respectively. By recurring to the tools of Finsler geometry, we finalize by writing the generalized gravitational field equations in curved phase space in the presence of matter sources, like particles and cosmic strings. As a result, both spacetime and momentum space are curved. We conclude with some remarks as to why curved momentum space should play an important role in quantum gravity.
It is shown that a careful study of the simplest family of generalized Clifford algebras (GCAs) a... more It is shown that a careful study of the simplest family of generalized Clifford algebras (GCAs) associated with the N-th root of unity in ddimensions leads to the following generalized anti-commutator with N entries {ei 1 , ei 2 , ei 3 ,. .. , ei N } = ei 1 ei 2 ei 3. .. ei N + permutations = N !ηi 1 i 2 ...i N e, where e is the unit element and all the N ! terms of the permutations appear with the same positive sign. The components of the rank-N metric are ηi 1 i 2 ...i N = 1, iff i1 = i2 = i3. .. = iN , and 0 otherwise. The range of the indices i1, i2,. .. iN is 1, 2,. .. , d. We proceed to explore the N-th norm extensions of the quadratic norm and write down a generalized Finsler-like arc-length based on the rank-N metric gµ 1 µ 2 ...µ N. We finalize by constructing the different expressions of the Dirac operators associated with the (Generalized) Clifford Spaces corresponding to these GCA's. Dirac operators are essential in the study of Spectral Geometry in Noncommutative Geometry after imposing the correspondence between the geodesic distance and the inverse of the Dirac operator (a fermion propagator). These generalized anti-commutators above are special types of an N-ary algebraic structure. We conclude with some remarks on N-ary algebras and their applications in Mathematics and Physics.
Starting with a brief review of Born Reciprocal (non-inertial) Relativity Theory (BRRT), it is sh... more Starting with a brief review of Born Reciprocal (non-inertial) Relativity Theory (BRRT), it is shown how massless photons in one frame of reference can appear massive in an acccelerated frame. An immediate application can be found in the behavior of in-falling/outgoing photons propagating in a black hole gravitational background where in-falling and outgoing photons from the point of view of an accelerated frame of reference (with respect to a static spherically symmetric Schwarzschild black hole, for example) will appear massive and subluminal. In view of these novel findings that massless particles can appear massive in accelerated frames, it should have many important consequences in cosmology (dark energy, dark matter problem) and QFT.
The $\kappa$-deformed exponential $ \exp_\kappa ( x) = \exp \left( { 1 \over \kappa } arcsinh ( ... more The $\kappa$-deformed exponential $ \exp_\kappa ( x) = \exp \left( { 1 \over \kappa } arcsinh ( \kappa x ) \right) $ studied by Kaniadakis \cite{K} allows to construct ``deformed" Lorentz transformations associated with the ordinary velocity boost rapidity parameter $ \xi$ and which can be recast in terms of ordinary Lorentz transformations (involving the ordinary exponential) but associated with a $\kappa$-deformed (modified) rapidity parameter $ \xi_\kappa = \xi f ( \kappa \xi ) $ given by a $ \xi$-dependent scaling of the original $ \xi$ rapidity parameter. It is shown that when $both$ the $ \kappa$ parameter and $ \xi \rightarrow \infty$, and the double scaling limit $ \xi { \ln ( 2 \kappa \xi ) \over \kappa \xi } = \infty \times 0 $ is finite and nonzero, it leads to a $finite$ value for the $\kappa$-deformed boost rapidity parameter $ \xi_{\kappa = \infty} = \xi_{ \infty} \not= \infty $, such that the $\kappa$-deformed velocity (in units of $ c = 1$) $ \tanh ( \xi_\kappa ) = v_\kappa < 1 $ is $less$ than the speed of light, and in turn, the Lorentz dilation factor $ \gamma ( v_\kappa) \not= \infty $ $no$ longer $blows$ up. Consequently, there is a $lower$ bound in the length $ L' = { L \over \gamma ( v_k ) } \not= 0 $ due to a $finite$ Lorentz length contraction. After imposing that the lower bound $ L' $ should not be smaller than the postulated minimum Planck scale one arrives at the length-scale-dependent relation $ arccosh ( { L \over L_P} ) = \xi_\infty > 0 $ that admits a physical interpretation analogous to the running of the physical couplings and masses with the energy scale in the Renormalization Group program in Quantum Field Theory.
It is shown how generalized Clifford algebras allows to construct the N-th root of N-order linear... more It is shown how generalized Clifford algebras allows to construct the N-th root of N-order linear differential equations involving massless and massive particles. Such generalized Dirac-like equations differ from the ones in the literature. Explicit solutions are found. We conclude with some remarks on pseudo-unitary algebras, modular arithmetic, modified Dirac equations, Octonions, and the Okubo algebra.
We review the basics of Thermal Relativity Theory and describe many of the novel physical consequ... more We review the basics of Thermal Relativity Theory and describe many of the novel physical consequences, including the $corrections$ to the Schwarzschild black hole entropy. One of the most salient results is the existence of a minimal area ${ L_P^2 \over 4 \pi} $, where $ L_P$ is the Planck length. The brief review paves the way to the study of the thermal relativistic analog of Lorentz transformations. A careful examination of these transformations imply that entropy is observer $dependent$ and that one must also include negative masses, entropy and temperature in the formalism. A review of the literature on negative masses, entropy and temperature follows. In special relativity one has the equivalence of mass and energy. While in thermal relativity one finds an equivalence of proper thermal mass $ {\cal M } $ (not to be confused with ordinary mass) and proper entropy $ {\bf s }$. We conclude with a brief description of Born's Reciprocal Relativity Theory (BRRT), based on a maximal proper-force, a maximal speed of light, inertial and non-inertial observers, and explain how to extend the formulation of thermal relativity described in this work to cotangent spaces.
The authors [1] have recently constructed models of nonextensive black hole Thermodynamics from a... more The authors [1] have recently constructed models of nonextensive black hole Thermodynamics from a generalized Wick's rotation procedure in the evaluation of the Euclidean path integral. We have explicitly shown in [6] how the Schwarzschild Black Hole Entropy (in all dimensions) emerges from truly point mass sources at r = 0 due to a non-vanishing scalar curvature involving the Dirac delta distribution. It is the density and anisotropic pressure components associated with the point mass delta function source at the origin r = 0 which furnish the Schwarzschild black hole entropy in all dimensions D ≥ 4 after evaluating the non-vanishing Euclidean Einstein-Hilbert action. In this work we generalize our construction of the Euclidean Einstein-Hilbert action by following the generalized Wick's rotation procedure of [1] in order to construct the nonextensive Schwarzschild black hole entropies in all dimensions. The first law of Thermodynamics is obeyed and when the nonextensivity parameter is λ < 0, the nonextensive entropy is finite at T = 0 despite that the Bekenstein-Hawking entropy S_BH (β = ∞) = S_BH (T = 0) = ∞ blows up violating the the third law of Thermodynamics.
International Journal of Modern Physics A, 2010
It is shown how a conformal gravity and U (4) × U (4) Yang–Mills grand unification model in four ... more It is shown how a conformal gravity and U (4) × U (4) Yang–Mills grand unification model in four dimensions can be attained from a Clifford gauge field theory in C-spaces (Clifford spaces) based on the (complex) Clifford Cl (4, C) algebra underlying a complexified four-dimensional space–time (eight real dimensions). Upon taking a real slice, and after symmetry breaking, it leads to ordinary gravity and the Standard Model in four real dimensions. A brief conclusion about the noncommutative star-product deformations of this Grand Unified Theory of gravity with the other forces of Nature is presented.
A continuation of the Born Reciprocal Relativity Theory (BRRT) program in phase space shows that ... more A continuation of the Born Reciprocal Relativity Theory (BRRT) program in phase space shows that a natural temperature-dependence of mass occurs after recurring to the Fulling-Davies-Unruh effect. The temperature dependence of the mass m(T) resemblances the energy-scale dependence of mass and other physical parameters in the renormalization (group) program of QFT. It is found in a special case that the effective photon mass is no longer zero, which may have far reaching consequences in the resolution of the dark matter problem. The Fulling-Davies-Unruh effect in a D = 1 + 1-dim spacetime is analyzed entirely from the perspective of BRRT, and we explain how it may be interpreted in terms of a linear superposition of an infinite number of states resulting from the action of the group U (1, 1) on the Lorentz non-invariant vacuum |\tilde 0⟩ of the relativistic oscillator studied by Bars [8].
Recently we have argued [1] that the noncommutativity of the spacetime coordinates is the answer ... more Recently we have argued [1] that the noncommutativity of the spacetime coordinates is the answer to the question : Why is area, mass, entropy quantized ? Furthermore, it casts light into a deep interplay among black hole entropy, discrete calculus, number theory, theory of partitions, random matrix theory, fuzzy spheres,. . .. We extend our previous construction of Schwarzschild black holes and derive the corrections to the Kerr-Newman temperature and black hole entropy, to all orders, from the discrete mass transitions taken place among different mass states. The mass spectrum for Kerr, Kerr-Newman, and Reissner-Nordstrom black holes is explicitly obtained which reduces to the Schwarzschild case when the angular momentum and charge is set to zero. One of the most salient features in the expansion of the modif ied temperature T = T + c_1 T/ N + c_2 T/N^2 +. .. is that it spells a correspondence between the loop expansion in QFT in powers of hbar, after setting hbar ↔ (1/N). N is the principal quantum number labeling the spectrum of mass states and which is given by N = l_3(l_3 + 2) − l_2(l_2 + 1) + l_1^2 , with l_3 ≥ l_2 ≥ |l_1| being the quantum numbers associated with the hyper-spherical harmonics of the three-sphere S^3. These results can be extended to higher dimensions. To finalize, we should add that the deviation from a full thermal spectrum and the corrections to the Hawking temperature might be relevant to the solution of the Black Hole Information paradox.
It is shown how a Noncommutative spacetime leads to an area, mass and entropy quantization condit... more It is shown how a Noncommutative spacetime leads to an area, mass and entropy quantization condition which allows to derive the Schwarzschild black hole entropy A/4G , the logarithmic corrections, and further corrections, from the discrete mass transitions taken place among different mass states in D = 4. The higher dimensional generalization of the results in D = 4 follow. The discretization of the entropy-mass relation S = S(M) leads to an entropy quantization of the form S = S(M_n) = n, and such that one may always assign n "bits" to the discrete entropy, and in doing so, make contact with quantum information. The physical applications of mass quantization, like the counting of states contributing to the black hole entropy, black hole evaporation, and the direct connection to the black holes-string correspondence [23] via the asymptotic behavior of the number of partitions of integers, follows. To conclude, it is shown how the recent large N Matrix model (fuzzy sphere) of [20] leads to very similar results for the black hole entropy as the physical model described in this work and which based on the discrete mass transitions originating from the noncommutativity of the spacetime coordinates.
We revisit the nonlinear Klein-Gordon-like equation that was proposed by us which capture how qua... more We revisit the nonlinear Klein-Gordon-like equation that was proposed by us which capture how quantum mechanical probability densities curve spacetime, and find an exact solution that may appear to be "trivial" but with important physical implications related to the physics of frozen stars and with Mach's principle. The nonlinear Klein-Gordon-like equation is essentially the static spherically symmetric relativistic analog of the Newton-Schrödinger equation. We finalize by studying the higher dimensional generalizations of the nonlinear Klein-Gordon-like equation and examine the relativistic Bohm-Poisson equation as yet another equation capturing the interplay between quantum mechanical probability densities and gravity.
A discrete Hopf fibration of S^15 over S^8 with S^7 (unit octonions) as fibers leads to a 16D Pol... more A discrete Hopf fibration of S^15 over S^8 with S^7 (unit octonions) as fibers leads to a 16D Polytope P_{16} with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Lambda_{16}. It is argued how a subsequent 2 − 1 mapping (projection) of P_{16} onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 2_{41} polytope in 8-dimensions, and such that one can capture the chain sequence of polytopes 2_{41}, 2_{31}, 2_{21}, 2_{11} in D = 8, 7, 6, 5 dimensions, leading, respectively, to the sequence of Coxeter groups E_8, E_7, E_6, SO(10) which are putative GUT group candidates. An embedding of the E8 ⊕ E8 and E8 ⊕ E8 ⊕ E8 lattice into the Barnes-Wall Λ_{16} and Leech Λ_{24} lattices, respectively, is explicitly shown. From the 16D lattice E8 ⊕ E8 one can generate two separate families of Elser-Sloane 4D quasicrystals (QC's) with H_4 (icosahedral) symmetry via the "cut-and-project" method from 8D to 4D in each separate E_8 lattice. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC's Q × Q spanning an 8D space. Similarly, from the 24D lattice E8 ⊕E8 ⊕E8 one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC's) Q × Q × Q with H_4 symmetry and spanning a 12D space.
One of the consequences of Fermat's last theorem is the existence of a countable infinite number ... more One of the consequences of Fermat's last theorem is the existence of a countable infinite number of rational points on the unit circle, which allows in turn, to find the rational points on the unit sphere via the inverse stereographic projection of the homothecies of the rational points on the unit circle. We proceed to iterate this process and obtain the rational points on the unit S^3 via the inverse stereographic projection of the homothecies of the rational points on the previous unit S^2. One may continue this iteration/recursion process ad infinitum in order to find the rational points on unit hyper-spheres of arbitrary dimension S^4 , S^5 , • • • , S^N. As an example, it is shown how to obtain the rational points of the unit S^{24} that is associated with the Leech lattice. The physical applications of our construction follow and one finds a direct relation among the N + 1 quantum states of a spin-N/2 particle and the rational points of a unit S^N hyper-sphere embedded in a flat Euclidean R^{N +1} space.
Recently we have shown how the Schwarzschild Black Hole Entropy in all dimensions emerges from tr... more Recently we have shown how the Schwarzschild Black Hole Entropy in all dimensions emerges from truly point mass sources at r = 0 due to a non-vanishing scalar curvature R involving the Dirac delta distribution in the computation of the Euclidean Einstein-Hilbert action. As usual, it is required to take the inverse Hawking temperature β as the length of the circle S^1_β obtained from a compactification of the Euclidean time in thermal field theory which results after a Wick rotation, it = τ , to imaginary time. In this work we extend our novel procedure to evaluate both the Reissner-Nordstrom and Kerr-Newman black hole entropy from truly charge spinning point mass sources.
It is explicitly shown how the Schwarzschild Black Hole Entropy (in all dimensions) emerges from ... more It is explicitly shown how the Schwarzschild Black Hole Entropy (in all dimensions) emerges from truly point mass sources at r = 0 due to a non-vanishing scalar curvature involving the Dirac delta distribution. It is the density and anisotropic pressure components associated with the point mass delta function source at the origin r = 0 which furnish the Schwarzschild black hole entropy in all dimensions D ≥ 4 after evaluating the Euclidean Einstein-Hilbert action. As usual, it is required to take the inverse Hawking temperature β_H as the length of the circle obtained from a compactification of the Euclidean time in thermal field theory which results after a Wick rotation, it = τ , to imaginary time. The appealing and salient result is that there is no need to introduce the Gibbons-Hawking-York boundary term in order to arrive at the black hole entropy because in our case one has that R is not 0. Furthermore, there is no need to introduce a complex integration contour to avoid the singularity as shown by Gibbons and Hawking. On the contrary, the source of the black hole entropy stems entirely from the scalar curvature singularity at the origin r = 0. We conclude by explaining how to generalize our construction to the Kerr-Newman metric by exploiting the Newman-Janis algorithm. The physical implications of this finding warrants further investigation since it suggests a profound connection between the notion of gravitational entropy and spacetime singularities.
A brief introduction of the Extended Relativity Theory in Clifford Spaces (Cspace) paves the way ... more A brief introduction of the Extended Relativity Theory in Clifford Spaces (Cspace) paves the way to the explicit construction of the generalized relativistic transformations of the Clifford multivector-valued coordinates in C-spaces. The most general transformations furnish a full mixing of the grades of the multivectorvalued coordinates. The transformations of the multivector-valued momenta follow leading to an invariant generalized mass M in C-spaces which differs from m. No longer the proper mass appearing in the relativistic dispersion relation E^2 −p^2 = m^2 remains invariant under the generalized relativistic transformations. It is argued how this finding might shed some light into the cosmological constant problem, dark energy, and dark matter. We finalize with some concluding remarks about extending these transformations to phase spaces and about Born reciprocal relativity. An appendix is included with the most general (anti) commutators of the Clifford algebra multivector generators.
We explore the construction of a generalized Dirac equation via the introduction of the notion of... more We explore the construction of a generalized Dirac equation via the introduction of the notion of Clifford-valued actions, and which was inspired by the work of [1], [2] on the De Donder-Weyl theory formulation of field theory. Crucial in this construction is the evaluation of the exponentials of multivectors associated with Clifford (hypercomplex) analysis. Exact matrix solutions (instead of spinors) of the generalized Dirac equation in D = 2, 3 spacetime dimensions were found. This formalism can be extended to curved spacetime backgrounds like it happens with the Schroedinger-Dirac equation. We conclude by proposing a wavefunctional equation governing the quantum dynamics of branes living in C-spaces (Clifford spaces), and which is based on the De Donder-Weyl Hamiltonian formulation of field theory.
After a brief introduction of Born's reciprocal relativity theory is presented, we review the con... more After a brief introduction of Born's reciprocal relativity theory is presented, we review the construction of the deformed Quaplectic group that is given by the semi-direct product of U (1, 3) with the deformed (noncommutative) Weyl-Heisenberg group corresponding to noncommutative fiber coordinates and momenta [X_a, X _b ] not 0; [P_a, P_b ] not 0. This construction leads to more general algebras given by a two-parameter family of deformations of the Quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks [X_a 1 a_2 •••a_n , X b_1 b_2 •••b_n ] not 0; [Pa_1 a_2 •••a_n , P b_1 b_2 •••b_n ] not 0. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed Quaplectic algebra. A solution is found for the exact analytical mapping of the non-commuting x_µ , p_µ operator variables (associated to an 8D curved phase space) to the canonical Y^A , Π^A operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the classical limit, with the embedding functions Y^A (x, p), Π^A (x, p) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric g_{µν} (x, p), the fiber metric of the vertical space h_{ab} (x, p), and the nonlinear connection N_{aµ}(x, p) associated with the 8D cotangent space of the 4D spacetime. Consequently, one has found a direct link between noncommutative curved phase spaces in lower dimensions to commutative flat phase spaces in higher dimensions.
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Papers by Carlos Castro Perelman