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Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events, as measured by observers situated at varying distances from a gravitating mass. The lower the gravitational potential (the closer the clock is to the source of gravitation), the slower time passes, speeding up as the gravitational ...
The faster the relative velocity, the greater the time dilation between them, with time slowing to a stop as one clock approaches the speed of light (299,792,458 m/s). In theory, time dilation would make it possible for passengers in a fast-moving vehicle to advance into the future in a short period of their own time.
The Shapiro time delay effect, or gravitational time delay effect, is one of the four classic Solar System tests of general relativity. Radar signals passing near a massive object take slightly longer to travel to a target and longer to return than they would if the mass of the object were not present.
As it is moving in S, we have γ>1, therefore its proper time is shorter with respect to time T. (For comparison's sake, another muon at rest on Earth can be considered, called muon-S. Therefore, its decay time in S is shorter than that of muon-S′, while it is longer in S′.) In S, muon-S′ has a longer decay time than muon-S.
That is, clocks at higher altitude tick faster than clocks on Earth's surface. This effect has been confirmed in many tests of general relativity, such as the Pound–Rebka experiment and Gravity Probe A. In the Hafele–Keating experiment, there was a slight increase in gravitational potential due to altitude that tended to speed the clocks ...
The twins meet at T=17.3 and τ=12. This is a different voyage than the one shown above, as both schemes take the same assumed total point-of-view time: T=12 (stay-at-home), resp τ=12 (ship), so the results of the calculated other-one's times must be different: τ=9.33 (ship), resp T=17.3 (stay at home). In the standard proper time formula
Objects are falling to the floor because the room is aboard a rocket in space, which is accelerating at 9.81 m/s 2, the standard gravity on Earth, and is far from any source of gravity. The objects are being pulled towards the floor by the same "inertial force" that presses the driver of an accelerating car into the back of their seat.
As a rule of thumb, for a constant acceleration at 1 g (Earth gravity), the journey time, as measured on Earth, will be the distance in light years to the destination, plus 1 year. This rule of thumb will give answers that are slightly shorter than the exact calculated answer, but reasonably accurate.