Radiation Beyond Four Space-Time Dimensions

We present a list of formulas describing classical radiation of the rank s tensor field from an accelerated point-like source in flat space-time of arbitrary even dimension d. This allows straightforward evaluating the total intensity and radiated momentum for any s and d algorithmically, by hands or with the help of a computer (e.g. with an attached MAPLE program). Practical application of formulas is limited, because, for s > 1, the energy-momentum tensor for the point-like source is not conserved. This usually means that one cannot neglect contributions to radiation from tensions of the forces that cause acceleration of the source. Traditionally radiation processes were considered as subjects of direct physical application and therefore were deeply investigated only in no more than d = 4 space-time dimensions, see [1]. Even for d = 2 and 3, where obvious applications exist, say, to sound waves in media, the theory remains badly represented in the literature. Only recently, after the strings-inspired multi-dimensional models [2] attracted increasing attention [3], some papers on multi-dimensional radiation began to emerge [4, 5, 6, 8, 7]. Of course, they are still too few to cover the field at the level of exhaustiveness, typical for the literature on 4d radiation. In this paper we make a step, which we think is necessary for a systematic study of physical effects. Namely, we provide a list of generic formulas, describing classical radiation for arbitrary dimension d and rank s of the radiated fields, what should help to reveal the underlying physical and mathematical structures. In particular, one can immediately read off the radiation damping force in higher dimensions from our results, for example by the method of [4]. To illustrate the main results of this paper, we use several Tables. We begin with Table I, summarizing the logic of radiation calculus. The left column is our main line, the right column enumerates related subjects which deserve deeper physical consideration and are left for a detailed presentation of radiation physics in a more sophisticated review article. According to Table I, in order to find radiation intensity one needs to perform the following chain of calculations: • Solve wave equation for a point-like source of the rank-s field, moving along a world line z(τ), 2Aμ1...μs(x) = ∮ uμ1 . . . uμsδ (d) ( x− z(τ) ) dτ, (1) and pick up the slowest-falling contribution at large distances. For even d it is given by a simple formula for the retarded Liénard-Wiechert potential: A μ1...μs = ( 1 (Ru) ∂τ ) d−4 2 uμ1 . . . uμs Ru and ∂μA rad μ1...μs = Rμ ( 1 (Ru) ∂τ ) d−2 2 uμ1 . . . uμs Ru (2) where uμ = ∂τzμ is source’s d-velocity, u 2 = 1, evaluated at the radiation-time moment t = z(τ) which is determined by the condition R = 0, where Rμ is the d-vector with components R μ ≡ x − z(τ). Here τ is source’s proper time. In particular, ∂τ ∂xμ = Rμ (Ru) (3) E-mail: [email protected]; [email protected] E-mail: [email protected]