does time pass slower in distant galaxies? if you could observe a planetary system at high redshift and observe the orbits of planets (or any other dynamical system driven by gravity), would the observation be consistent with a lower gravity instead of a slower passage of time? what about atomic processes like chemical reactions or radiative processes?

bonus question: can you show that for a real-valued matrix $A$ the matrix exponential $\exp(A)$ transforms according to

\begin{equation}

\exp(R^{-1}AR) = R^{-1}\exp(A)R

\end{equation}

with any invertible matrix $R$?

bonus question: you can expand the matrix exponential in a series

ReplyDelete\begin{equation}

\exp(R^{-1}AR)

= \sum_i\frac{(R^{-1}AR)^i}{i!}

= R^{-1}\:\:\sum_i\frac{A^i}{i!}\:\:R

= R^{-1}\exp(A)R

\end{equation}

because of the associativity of the matrix product, $(R^{-1}AR)^i = R^{-1}AR\ldots R^{-1}AR = R^{-1}A\ldots AR = R^{-1}A^iR$.

we see processes in distant galaxies in slow motion because the arrival times of photons gets stretched by the Hubble expansion. in classical planetary systems a rescaling of the gravitational constant is completely degenerate with choosing a different mass of the central body, and the orbit only has to obey Kepler's third law, with a possibly different constant of proportionality. so yes, if we didn't have spectroscopy and could not measure redshifts, one could explain the slowly moving planets by lower gravity.

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