# On Clifford Space Relativity, Black Hole Entropy, Rainbow Metrics, Generalized Dispersion and Uncertainty Relations

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DOI: 10.1007/s10701-014-9825-x

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- Castro, C. Found Phys (2014) 44: 990. doi:10.1007/s10701-014-9825-x

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## Abstract

An analysis of some of the applications of Clifford space relativity to the physics behind the modified black hole entropy-area relations, rainbow metrics, generalized dispersion and minimal length stringy uncertainty relations is presented.

### Keywords

Clifford algebras Extended relativity in Clifford spaces String theory Doubly special relativity Rainbow metrics Black hole entropy Noncommutative geometry Quantum Clifford–Hopf algebras## 1 Introduction: Novel Consequences of Clifford Space Relativity

In the past years, the extended relativity theory in \(C\)-spaces (Clifford spaces) and Clifford-phase spaces were developed [1, 2]. The extended relativity theory in \(C\)-spaces is a natural extension of the ordinary relativity theory whose generalized coordinates are Clifford polyvector-valued quantities which incorporate the lines, areas, volumes, and hyper-volumes degrees of freedom associated with the collective dynamics of particles, strings, membranes, p-branes (closed p-branes) moving in a D-dimensional target spacetime background. \(C\)-space relativity permits to study the dynamics of all (closed) \(p\)-branes, for different values of \(p = 0, 1, 2,\ldots ,\) on a unified footing. Our theory has \(2\) fundamental parameters : the speed of a light \(c\) and a length scale which can be set equal to the Planck length. The role of “photons” in \(C\)-space is played by \(tensionless\)\(p\)-branes since a \(p\)-brane is the extended object version of a point particle (\(p = 0\)), and the tension of a \(p\)-brane (mass per unit length, area, volume, \(\ldots \)) is physically what corresponds to the mass of a point particle [1].

An extensive review of the extended relativity theory in \(C\)-spaces can be found in [1]. The polyvector valued coordinates \( x^\mu , x^{\mu _1 \mu _2 }, x^{\mu _1 \mu _2 \mu _3},\ldots \) are now linked to the basis vectors generators \( \gamma ^\mu \), bi-vectors generators \( \gamma _\mu \wedge \gamma _\nu \), tri-vectors generators, \( \gamma _{\mu _1} \wedge \gamma _{\mu _2} \wedge \gamma _{\mu _3},\ldots \) of the Clifford algebra, \( \{ \gamma _a, \gamma _b\} = 2 g_{ab} \mathbf{1} \), including the Clifford algebra unit element (associated to a scalar coordinate). These polyvector valued coordinates can be interpreted as the quenched-degrees of freedom of an ensemble of \(p\)-loops associated with the dynamics of closed \(p\)-branes, for \(p = 0, 1, 2, \ldots , D -1\), embedded in a target \(D\)-dimensional spacetime background.

*polyvector*or

*Clifford aggregate*which we now write in the form

*holographic coordinates*, since they describe the holographic projections of 1-loops, 2-loops, 3-loops\(,\ldots ,\) onto the coordinate planes . By \(p\)-loop we mean a closed \(p\)-brane; in particular, a 1-loop is closed string. In order to avoid using the powers of the Planck scale length parameter \(L_p\) in the expansion of the polyvector \(X\) (in order to match units) we can set it to unity to simplify matters. In a \(flat\)\(C\)-space the basis vectors \(E^A, E_A\) are \(constants\). In a \(curved\)\(C\)-space this is no longer true. Each \(E^A, E_A\) is a function of the \(C\)-space coordinates

Recently, novel physical consequences of the extended relativity theory in \(C\)-spaces were explored in [3, 4]. The latter theory provides a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the doubly special relativity (DSR) framework. Furthermore, an elegant \(nonlinear\) momentum-addition law was derived in order to tackle the “soccer-ball” problem in DSR. Neither derivation in \(C\)-spaces requires a \(curved\) momentum space nor a deformation of the Lorentz algebra. While the constant (energy-independent) speed of photon propagation is always compatible with the generalized photon dispersion relations in \(C\)-spaces, another important consequence was that the generalized \(C\)-space photon dispersion relations allowed also for energy-dependent speeds of propagation while still \(retaining\) the Lorentz symmetry in ordinary spacetimes, while breaking the \(extended\) Lorentz symmetry in \(C\)-spaces. This does \(not\) occur in DSR nor in other approaches, like the presence of quantum spacetime foam.

We learnt from special relativity that the concept of simultaneity is also relative. By the same token, we have shown in [3, 4] that the concept of spacetime locality is \(relative\) due to the \(mixing\) of area-bivector coordinates with spacetime vector coordinates under generalized Lorentz transformations in \(C\)-space. In the most general case, there will be mixing of all polyvector valued coordinates. This was the motivation to build a unified theory of all extended objects, \(p\)-branes, for all values of \(p\) subject to the condition \( p+1 = D\).

In [5, 6, 7] we explored the many novel physical consequences of Born’s reciprocal relativity theory [8, 9, 10, 11, 12, 13] in flat phase-space and generalized the theory to the curved phase-space scenario. We provided six specific novel physical results resulting from Born’s reciprocal relativity and which are \(not\) present in special relativity. These were : momentum-dependent time delay in the emission and detection of photons; energy-dependent notion of locality; superluminal behavior; relative rotation of photon trajectories due to the aberration of light; invariance of areas-cells in phase-space and modified dispersion relations. We finalized by constructing a Born reciprocal general relativity theory in curved phase-spaces which required the introduction of a complex Hermitian metric, torsion and nonmetricity.

We should emphasize that *no* spacetime foam was introduced, nor Lorentz invariance was broken, in order to explain the time delay in the photon emission/arrival. In the conventional approaches of DSR where there is a Lorentz invariance breakdown [14, 15, 16], a longer wavelength photon (lower energy) experiences a smoother spacetime than a shorter wavelength photon (higher energy) because the higher energy photon experiences more of the graininess/foamy structure of spacetime at shorter scales. Consequently, the less energetic photons will move faster (less impeded) than the higher energetic ones and will arrive at earlier times.

However, in our case above [5, 6, 7] the time delay is entirely due to the very nature of Born’s reciprocal relativity when one looks at pure acceleration (force) boosts transformations of the phase space coordinates in *flat* phase-space. No *curved* momentum space is required as it happens in [14, 15, 16]. The time delay condition in Born’s reciprocal relativity theory implied also that higher momentum (higher energy) photons will take longer to arrive than the lower momentum (lower energy) ones.

Superluminal particles were studied within the framework of the extended relativity theory in *C*-spaces in [8]. In the simplest scenario, it was found that it is the contribution of the Clifford scalar component *P* of the poly-vector-valued momentum **P** which is responsible for the superluminal behavior in ordinary spacetime due to the fact that the effective mass \( \sqrt{ \mathcal{M}^2 - P^2 } \) can be imaginary (tachyonic). However from the point of view of \(C\)-space there is no superluminal behaviour (tachyonic) because the true physical mass still obeys \( \mathcal{M}^2 > 0 \). As discussed in detailed by [1, 17, 18] one can have tachyonic (superluminal) behavior in ordinary spacetime while having non-tachyonic behavior in *C*-space. Hence from the *C*-space point of view there is no violation of causality nor the Clifford-extended Lorentz symmetry.

The addition law of areal velocities and a minimal length interpretation \( L \) was recently studied in [3, 4]. The argument relied entirely on the physics behind the extended notion of Lorentz transformations in \( C\)-space, and *does not* invoke quantum gravity arguments nor quantum group deformations of Lorentz/Poincare algebras. The physics of the extended relativity theory in *C*-spaces requires the introduction of the speed of light and a minimal scale. In [2] we have shown how the construction of an extended relativity theory in Clifford *Phase* spaces requires the introduction of a \(maximal\) scale which can be identified with the Hubble scale and leads to modifications of gravity at the Planck/Hubble scales. Born’s reciprocal relativity demands that a minimal length corresponds to a minimal momentum that can be set to be \(p_{min} = \hbar /R_{Hubble}\). For full details we refer to [2].

Despite the fact that the length parameter \(L\) (which must be introduced in the \(C\)-space interval in Eq. (1.2) in order to match units) has the physical interpretation of a \(minimal\) length, this does \(not\) mean that the spatial separation between two events in \(C\)-space cannot be \(smaller\) than \(L\). The Planck scale minimal length argument is mainly associated with quantum mechanics and black hole physics. The energy involved in the physical measurement process to localize a Planck mass particle, within Planck scale resolutions, becomes very large and such that a black hole forms enclosing the particle behind the black hole horizon. Since one does not have physical access to the black hole interior one cannot probe scales beyond the Planck scale. We shall set aside for the moment the current firewall controversy of black holes.

Recently, we improved our earlier work in [19] and derived the minimal length string/membrane uncertainty relations by imposing momentum slices in flat Clifford spaces [20]. The Jacobi identities associated with the modified Weyl–Heisenberg algebra require noncommuting spacetime coordinates, but commuting momenta, and which is compatible with the notion of curved momentum space. The purpose of this work is mainly to follow a different approach than the one taken in [20] by noticing that rainbow metrics [21, 22] are a natural consequence of taking momentum slices in \(C\)-spaces. Generalized dispersion and uncertainty relations are found in addition to modified black hole area-entropy relations.

## 2 On Rainbow Metrics and Generalized Dispersion and Uncertainty Relations from Clifford Spaces

### 2.1 Clifford Space Relativity Induces Generalized Dispersion and Uncertainty Relations

In this section we shall provide a \(different\) derivation of the generalized uncertainty relations than the one described in [20]. Our derivation in this work is based on the concept of rainbow metrics [21, 22].

The generalization of the Weyl–Heisenberg algebra to \(C\)-spaces and involving polyvector-valued coordinates and momenta (in natural units \(\hbar =1\)) is [1] \( [X_A, P_B ] = i G_{AB} \) and does not lead to minimal uncertainty conditions for \( \Delta X_A\). To obtain the minimal length stringy uncertainty relations in ordinary spacetimes requires more work. It involves taking polymomentum slices through \( C\)-space. This is the subject of this section.

### 2.2 Jacobi Identities and Noncommutative Spacetime

### 2.3 Lorentz Invariant Case

If one were to set \( \pi = 0\) in (2.40) it leads to a quartic algebraic equation for \( p^2 \) and that will fix the numerical values of \( p^2\) given by the four roots of the algebraic equation. The four roots are themselves functions of \( \mathcal{M}^2 \) and the parameters \( \lambda _1, \lambda _2, \lambda _3\). The rainbow function squared \( f^2 ( p^2) \) will have \(fixed\) numerical values instead of being a variable function and hence the rainbow metric \( g_{ \mu \nu } \) will be just trivially proportional to the Minkowski metric \( \eta _{\mu \nu } \) and will not modify the Weyl–Heisenberg algebra since one could reabsorb the constant of proportionality into \( \hbar \). For this reason one must retain \( \pi \) and \( \sigma \) in Eqs. (2.40) and (2.42).

## 3 On Rainbow Metrics and Modified Black Hole Entropy-Area Relation

This section is devoted to the specific physical applications associated to the rainbow metric modifications of the Schwarzschild black hole and its thermodynamical implications due to the modified Hawking temperature and entropy-area relations.

Recently, instead of recurring to momentum-dependent rainbow metrics, \(radial\) corrections to the Schwarzschild metric, necessary to reproduce the Hawking temperature derived from a generalized uncertainty principle (GUP), were found by [35] so that the GUP deformation parameter \(\beta \) is directly linked to the deformation of the metric. The problem with this result is that \( \beta < 0 \) turns out to be \(negative\) and this yields an \(imaginary\) value for the minimal uncertainty \( (\Delta x)_{min} \) in Eq. (2.28). To cure this problem, one could include higher order corrections to the modified uncertainty relation as displayed by Eq. (2.29) and involving more than one deformation parameter.

## 4 Concluding Remarks

To summarize this work, we have shown how Clifford space relativity provides generalized dispersion and modified uncertainty relations after imposing constraints in the polymomentum mass shell condition (2.1) given by Eqs. (2.2) and (2.41), in the non-Lorentz and Lorentz invariant case, respectively. The modified dispersion relation can be recast in the form displayed by Eq. (2.15) after imposing certain conditions on the numerical parameters. Two approaches were taken in Sect. 2.1 to obtain the modified uncertainty relations. One was based on a constraint among \( \pi ^2 \) and \( |{ \vec p }|^2\) provided by Eq. (2.7). The second approach relied on the notion of the rainbow metric given by Eq. (2.12) and originating from Eqs. (2.3) and (2.9). Specific solutions for the momentum dependence of the rainbow metrics were derived in Eqs. (2.23c) and (2.45) for the non-Lorentz and Lorentz invariant cases, respectively. The Jacobi identities in Sect. 2.2 led to a noncommutativity of the spacetime coordinates displayed in Eq. (2.34). Finally in Sect. 3 we have analyzed further the rainbow metric modifications of the Schwarzschild black hole and its thermodynamical implications due to the modified Hawking temperature and entropy-area relations. We arrived at the relation in Eq. (3.12), to leading order, constraining the ratio of the rainbow functions after equating the entropy integrands in Eqs. (3.4) and (3.10).

We conclude with some final remarks. Related to the minimal length uncertainty in Eq. (2.28) one should mention that the theory of scale relativity proposed by Nottale [36, 37] is based on a minimal observational length-scale, the Planck scale, as there is in special relativity a maximum speed, the speed of light, and deserves to be looked within the Clifford algebraic perspective. In the quantization program of gravity a key role must be played by quantum Clifford–Hopf algebras since the latter \(q\)-Clifford algebras naturally contain the \( \kappa \)-deformed Poincare algebras [38, 39, 40], which are essential ingredients in the formulation of DSR within the context of Noncommutative spaces. The Minkowski spacetime quantum Clifford algebra structure associated with the conformal group and the Clifford–Hopf alternative \(\kappa \)-deformed quantum Poincare algebra was investigated [41]. The resulting algebra is equivalent to the deformed anti-de Sitter algebra \(U_q(so(3,2))\), when the associated Clifford–Hopf algebra is taken into account, together with the associated quantum Clifford algebra and a (not braided) deformation of the periodicity Atiyah–Bott–Shapiro theorem [42]. In future work we shall address the fractal nature of quantum spacetime [36, 37] within the framework of quantum Clifford algebras and scale relativity.

## Acknowledgments

We are indebted to M. Bowers for assistance and to the referees for their many suggestions to improve this work.