Hawking radiation from a (4+n)-dimensional black hole: exact results for the Schwarzschild phase

Abstract

We start our analysis by deriving a master equation that describes the motion of a field with arbitrary spin s on a 3-brane embedded in a non-rotating, uncharged (4 + n)-dimensional black hole background. By numerical analysis, we derive exact results for the greybody factors and emission rates for scalars, fermions and gauge bosons emitted directly on the brane, for all energy regimes and for an arbitrary number n of extra dimensions. The relative emissivities on the brane for different types of particles are computed and their dependence on the dimensionality of spacetime is demonstrated -we therefore conclude that both the amount and the type of radiation emitted can be used for the determination of n if the Hawking radiation from these black holes is observed. The emission of scalar modes in the bulk from the same black holes is also studied and the relative bulk-to-brane energy emissivity is accurately computed. We demonstrate that this quantity varies considerably with n but always remains smaller than unity -this provides firm support to earlier arguments made by Emparan, Horowitz and Myers.The above relation involves the volume of the extra dimensions, V ∼ R n , under the assumption that R is the common size of all n extra compact dimensions. Therefore, if the volume of the internal space is large (i.e. if R ≫ ℓ P l , where ℓ P l = 10 −33 cm is the Planck length) then M * can be substantially lower than M P l .In the regime r ≪ R, the extra dimensions 'open up' and gravity becomes strong. Hence, Newton's law for the gravitational interactions in this regime is modified, with the gravitational potential assuming a 1/r n+1 dependence on the radial separation between two massive particles. Experiments which measure the gravitational inversesquare law at short scales can provide limits on the size of the extra dimensions, or equivalently on the value of the fundamental scale M * ; for n = 2 such measurements give M * > 3.5 TeV [4] (the n = 1 case has already been excluded by astronomical data). Since gravitons can propagate both in the bulk and on the brane, massive Kaluza-Klein (KK) graviton states can modify both the cross sections of Standard Model particle interactions and astrophysical or cosmological processes. The absence of signatures of production of either real or virtual KK gravitons at colliders puts a relatively weak lower limit on M * -from 1.45 TeV (for n = 2) to 0.6 TeV (n = 6) [5]. Much more stringent constraints arise if one considers astrophysical or cosmological processes; ignoring the systematic errors, these constraints exclude by far even the n = 3 case, while allowing models with M *

On Effective Potential in Tortoise Coordinate

“…where the functions f (r) and θ(r) are uniquely determined such that one be able to bring the equation of motion of φ field in Schrodinger-like form [2] …”

“…Spinor field representation in three dimension has two components and the coupled equation of motion for these fields is given by Dirac equation in curved background as [2] (−iγ a e µ a D µ + m)Ψ(t, θ, r) = 0,…”

“…This coordinate system is defined such that one be able to write down the equation of motion of a scalar field, for example, like Schrodinger equation [2]. Finding and studying physics in this coordinate system has several advantage.…”

“…For example, after obtaining that, one may be able, by solving the Schrodinger equation, to quantize the system exactly and find the spectrum and Hilbert space of the theory. Instead of this, because of the effects of the boundary conditions on the dynamics of the fields, one may interest in the asymptotic behavior of the field dynamics, i.e at far infinity and near the horizon 2 [2,3]. In Tortoise coordinate, this feature can also be addressed by evaluating the effective potential at boundary of space-time.…”

Manifestations of string theory in astrophysical data and at the LHC

“…the greybody factor) [109]. The SM fields live on a 3-brane Indeed, as the total angular momentum number of the emitted field increases, σ s (ω) is rapidly suppressed [110,111,112,113]. In the low energy limit, ω r ≪ 1, higher-order terms are suppressed by a factor of 3(ω r) −2 for fermions and by a factor of 25(ω r) −2 for gauge bosons.…”