Correction due to the finite speed of light in absolute gravimeters

Abstract

Abstract. Correction due to finite speed of light is among the most inconsistent ones in absolute gravimetry. Formulas reported by different authors yield corrections scattered up to 8 µGal with no obvious reasons. The problem, though noted before, has never been studied, and nowadays the correction is rather postulated than rigorously proven. In this paper we make an attempt to revise the subject. Like other authors, we use physical models based on signal delays and the Doppler effect, however, in implementing the models we additionally introduce two scales of time associated with moving and resting reflectors, derive a set of rules to switch between the scales, and establish the equivalence of trajectory distortions as obtained from either time delay or distance progression. The obtained results enabled us to produce accurate correction formulas for different types of instruments, and to explain the differences in the results obtained by other authors. We found that the correction derived from the Doppler effect is accountable only for 2 3 of the total correction due to finite speed of light, if no signal delays are considered. Another major source of inconsistency was found in the tacit use of simplified trajectory models.

Dynamic phase shift within a falling glass cube is negligible: comment on ‘Relativistic theory of the falling retroreflector gravimeter’

“…which agrees with the results of [2][3][4] 3 . Then the correction for the case considered in [1] (the beamsplitter is below the falling cube) is obtained from (7) as…”

“…The last component Z γ/c (T) with the factor γ/c contains the remaining negligible terms. The speed-of-light perturbation component (4) agrees with that shown in [2], in equation (13) 2 . Taking the second derivative of (4), the perturbing acceleration yields…”

“…(14) Least-squares adjustment of the model (14) to the displacement (13) without L returns unbiased estimates 6 :…”

No correction for the light propagation within the cube: comment on ‘Relativistic theory of the falling retroreflector gravimeter’

“…As seen in (4), the equation (2) has all necessary time delays (terms with 1/c 2 ) to account for the finite speed of light, even though the delays are not treated in [1] explicitly. We agree with [1] that using formula (2) to get the acceleration requires no additional 'speed of light' correction. The formulae (4) confirms that the 'speed of light' perturbation is proportional to the reflector's velocity with coefficient 3g/c.…”

Reply to ‘Dynamic phase shift within a falling glass cube is negligible: comment on ‘Relativistic theory of the falling cube gravimeter’’

“…On the other hand, user manuals for such gravimeters give the equivalent of the CM position Z cm (T) for the model of free fall [4], the constants being 'free parameters providing the best estimates for initial position, velocity and gravity'. In many references, Z 0 is regarded as an adjustable parameter [5][6][7]. Measurements can provide a good estimate for Z 0 but allowance is usually made for variations in the conditions of each individual drop.…”