Final answer:
Sending photons in opposite directions in a closed universe can result in different proper time before we see them again due to the expansion of the universe and the redshift experienced by the photons.
Step-by-step explanation:
In a closed universe, sending photons in opposite directions can result in different proper time before we see them again. This is because the universe is expanding, causing the photons to experience redshift. Redshift refers to the stretching of the photon's wavelength as the universe expands, which leads to a decrease in energy.
As a consequence, the energy of the photon is not strictly conserved over cosmological timescales. The proper time experienced by the photons will be different as they travel through the expanding universe, resulting in different elapsed times before they are observed again.