The warping of time: synopsis
The warping of time by gravity is evidenced by the fact that clocks on planes high above the ground run faster than those subject to stronger gravity at ground level, although in our earthly environment the effects of gravity in these circumstances are so miniscule as to pass virtually unnoticed. Clocks in motion move more slowly than those which are stationary, and actually stop at the speed of light[1]. From our limited three-dimensional perspective, we have a myopic view of reality and split space and time into two different entities[2].
Energy being equal to mass, an object approaching the speed of light will increase in mass to such an extent that the speed of light cannot be exceeded. Gravity slows the passage of time - clocks high on a mountain will run faster than those at ground level where gravity is stronger though the difference on earth will again pass virtually unnoticed[2.5]. Time is a concept relative to the presence of mass, the example of which we are most familiar being the passage of time on our own planet. Stronger gravitational fields such as those just outside a black hole cause the flow of time to slow even further. Since, in Einsteinian mechanics, gravitation curves spacetime in a way such that motion occurs without an applied force, it is as natural for a free-falling body to fall towards the mass as it is to move forward in time[3].
Perhaps a more accurate way of stating these principles is to say that gravity does not actually cause time to run differently in different places (faster far from the earth than near it), and that there are two separate phenomena, gravity and time, the former affecting the latter, but that the phenomena we usually ascribe to gravity are actually caused by time flowing unequably from place to place.[4]
Source: Lewis Carroll Epstein, “Relativity Visualized”, in http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_spacetime.html
George Musser also explains[5], the morphing of space and time govern the motion of any falling object . "Baseballs , wineglasses, expensive smartphones: things that slip out of your hand accelerate towards the floor because Earth's mass warps time. (The warping of space plays only a minor role in these cases). 'Down' is defined by the direction in which time passes more slowly. Clocks at sea level tick more slowly than clocks on the summit of Denali; a watch strapped to your ankle will fall behind one on your wrist. In human terms, the deviations are small - parts in a trillion at most - but enough to account for the rate at which falling objects pick up speed". And then this: "When you see an apple fall from a tree, you are watching it roll across the contours of time".
[1] In the somewhat different context of wormholes (hypothetical tunnels through space), Brian Greene gives an example of a voyage to the Andromeda galaxy and back at 99.9 (repeater) % of the speed of light occupying four hours during which some 6 million years elapsed on earth: Greene (2005), 463-4.
[2] Marcelo Gleiser, Imperfect Creation, 68.
[2.5] http://www.dailymail.co.uk/sciencetech/article-1314656/Scientists-prove-time-really-does-pass-quicker-higher-altitude.html
[3] http://www.physicsforums.com/showthread.php?t=563435
[4] W.G.Unruh, ‘Time, Gravity and Quantum Mechanics at:
http://www.physicsforums.com/showthread.php?t=165511
[5] "Where is here?", Scientific American", November 2015, 60 at 62: a preview from his forthcoming book "Spooky action at a distance".