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We compare the redshifts of neutral carbon and carbon monoxide in theredshifted sources in which the fine structure transition of neutral carbon,[CI], has been detected, in order to measure space-time variation of thefundamental constants. Comparison with the CO rotational lines measures givesthe same combination of constants obtained from the comparison fine structureline of singly ionised carbon, [CII]. However, neutral carbon has the distinctadvantage that it may be spatially coincident with the carbon monoxide, whereas[CII] could be located in the diffuse medium between molecular clouds, and soany comparison with CO could be dominated by intrinsic velocity differences.Using [CI], we obtain a mean variation of dF/F = (-3.6 +/- 8.5) x 10^-5, over z= 2.3 - 4.1, for the eight [CI] systems, which degrades to (-1.5+/- 11) x10^-5, over z = 2.3 - 6.4 when the two [CII] systems are included. That is,zero variation over look-back times of 10.8-12.8 Gyr. However, the latestoptical results indicate a spatial variation in alpha, which describes a dipoleand we see the same direction in dF/F. This trend is, however, due to a singlesource for which the [CI] spectrum is of poor quality. This also applies to oneof the two [CII] spectra previously used to find a zero variation inalpha^2/mu. Quantifying this, we find an anti-correlation between |dF/F| andthe quality of the carbon detection, as measured by the spectral resolution,indicating that the typical values of >50 km/s, used to obtain a detection, aretoo coarse to reliably measure changes in the constants. From the fluxes of theknown z > 1 CO systems, we predict that current instruments are incapable ofthe sensitivities required to measure changes in the constants through thecomparison of CO and carbon lines. We therefore discuss in detail the use ofALMA for such an undertaking ... ABRIDGED

Measuring space-time variation of the fundamental constants with redshifted submillimetre transitions of neutral carbon