Definition of uniform acceleration and its conformai invariance

Abstract

The usual equation defining uniform acceleration of a particle in special relativity, namely (1−u2)u′’+3(uu′)u′=0 whereu is the velocity and primes denote differentiation with respect to time, is shown to be equivalent to (i) vΜ=KvΜ, Μ=0, 1, 2, 3 wherevΜ is the four-velocity, dots denote differentiation with regard to the proper time, Τ, andK is a costant. On integration this gives (ii)vΜ=αΜ exp [λΤ]+ΒΜ exp [-λΤ] and (iii)xΜ-ξΜ= (1/λ)[αΜ exp [λΤ]-ΒΜ exp [-λΤ]] where λ=√K and αΜ,ΒΜ ξΜ are integration constants. Three-dimensional forms of these equations are given. The invariance of these equations is examined and it is shown that under the infinitesimal conformai transformations of the coordinates δξΜ=(axxΜ)-1/2x2aΜ, [(ax)=aαxα], they are invariant providedK, ξΜ, αΜ,gaΜ andΒΜ transform suitably, namely according to δK=-2K(aξ), δξΜ=(aξ)ξΜ-1/2 ξaΜ+(1/K(1/2aΜ-2CΜ), δαΜ=(αa)ξΜ-(αξ)aΜ δΒΜ=(Βa)ξ-(Μξ)aΜCΒ = (aΒ)αΜ+(aα)ΒΜ. Since the transformation law for ξΜ is inhomogeneous, it follows that the constants ξΜ cannot be taken zero, nor are eitherK or αΜ orΒΜ absolute constants. It is interesting that the eq. (i) for vΜ is invariant only if we consider it together with its integral (ii); in turn, (ii) is invariant only if we consider it together with its integral (iii). Taken by themselves alone, neither (i) nor (ii) are conformally invariant, but (iii) is.