If not, why is my idea below not possible?
The general theory of relativity tells us that time is slowed down by gravitational fields. We bounced radar signals off planets further from the sun, and ones nearer to it, and found that the light bounced off nearer ones was red-shifted, due to passing through stronger gravitational fields.
The universe has a great deal of mass in it. Although far from concentrated sources of mass, space-time is 'flatter', it should still be true that everywhere within the universe is within its deep gravitational 'well'. As a consequence, all light travelling through space should be slowed in time and red-shifted proportional to how far it travels - no?
