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Radio sources with steep spectra are preferentially associated with the mostdistant galaxies, the $\alpha-z$ relation, but the reason for this relation isan open question. The spatial distribution of spectra in high-z radio sourcescan be used to study this relation, and low-frequency observations areparticularly important in understanding the particle acceleration and injectionmechanisms. However, the small angular sizes of high-z sources together withthe inherently low resolution of low-frequency radio telescopes until now hasprevented high angular resolution low-frequency observations of distantobjects. Here we present subarcsecond observations of a $z = 2.4$ radio galaxyat frequencies between $121$ MHz and $166$ MHz. We measure the spatialdistribution of spectra, and discuss the implications for models of the$\alpha-z$ relation. We targeted 4C 43.15 with the High Band Antennas (HBAs) ofthe \textit{International LOFAR Telescope} (ILT) with a range of baselines upto $1300\ \mathrm{km}$. At the central frequency of $143$ MHz we achieve anangular resolution of $\sim 0.3''$. By complementing our data with archival\textit{Very Large Array} (VLA) data we study the spectral index distributionacross 4C 43.15 between $55\ \mathrm{MHz}$ and $8.4\ \mathrm{GHz}$ atresolutions of $0.4''$ and $0.9''$. With a magnetic field strength of $B = 5.2$nT and fitted injection indices of $\alpha^\mathrm{north}_\mathrm{inj} = -0.8$and $\alpha^\mathrm{south}_\mathrm{inj} = -0.6$, fitting a Tribble spectralageing model results in a spectral age of $\tau_\mathrm{spec} = 1.1 \pm 0.1$Myr. We conclude that our data on 4C 43.15 indicates that inverse Comptonlosses could become comparable to or exceed synchrotron losses at higherredshifts and that inverse Compton losses could be a viable explanation for the$\alpha-z$ relation.

High-resolution international LOFAR observations of 4C~43.15 -- Spectral ages and injection indices in a high-z radio galaxy