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Gravitational force
The thing that is relevant for our purpose is time running faster in space at direction in which the object accelerates and slower in the opposite direction. An accelerated observer is seeing clocks ticking faster than normal in the direction of acceleration and slower than normal in opposite direction. The difference is proportional to the acceleration and the distance to the clock. It is rather easy to figure out why it has to be so:
Let's imagine ourselves sitting at the rear end of a long room comparing a clock on the front-end wall with our wristwatch. If the room is at rest or moves with a constant speed (in relation to something that does not accelerate as verified by an accelerometer fixed to it) the clock on the wall has to be set a little ahead to look synchronized with our wristwatch. It is because the light from clock on the wall needs time to reach us across the room. When the room starts accelerating, we will see images of the clock getting to us a little earlier than when the room moved with constant speed, because we encounter those images of the clock moving against them with higher and higher speed. This way the clock looks running ahead of our wristwatch proportionally to the acceleration and to the distance to the clock.
When we stop accelerating the clock on the wall will be running at the same rate as our wristwatch again, but it won't be in sync anymore because we've seen it running faster when we were accelerating and it can't just jump back to "right time" when we stopped accelerating. It will show more ticks than our wristwatch. It shows that during the acceleration the time was running faster at the place where that clock is (at the wall in front of us).
This is one of facts that people have hard time to believe. It turns out to be a physical fact: rate of time in respect to us is not fixed but depends on various conditions. E.g. it depends on our acceleration and the distance to the place when the rate of time is measured.
The opposite will happen to the clock behind us. It will show fewer ticks, which confirms that it has been running slower. Or that time has been running slower there where this clock is.
Those effects have been noticed by the physicists before Einstein but they were considered to be too mysterious to think about since it was not considered possible that time may run faster or slower than at its usual rate. It was left for future generations to explain, which Einstein did by demonstrating that if the speed of light is always the same then the time actually has to run faster or slower depending on various physical conditions. It was a simple explanation and a basis of Einstein's special relativity that deals with systems that move with constant velocity.
Then, when Einstein started considering also accelerated movements this variable time rate became a basis for a new theory of gravity called general relativity.
All the above is the explanation of the second paradox in the twin paradox. This second part was that the if rate of change of velocity was causing one twin aging faster than the other then the velocity of one twin with respect to the other was the same as the other in respect to the first and so were their mutual "accelerations". So it is not the rate of change of velocity with respect to something, that causes those effects, but real acceleration that is a physical fact measured by an accelerometer regardless whether it is related to any velocity with respect to any reference point.
The traveling twin who flies to a distant star has to accelerate toward the earth when she starts her return trip and her accelerometer will show it. During this acceleration caused by the change of direction of the movement from away from the earth to towards the earth the time on the earth runs much faster in relation to the accelerating twin. It runs faster proportionally to the change in velocity and to the distance to the earth. This faster running time has never a chance to wind back so on her return to the earth the traveling twin meets her twin that is now older than the traveling twin.
It would be the same if the traveling twin were just sitting on a more massive planet then the earth (while her twin was sitting on the earth).
While sitting on a planet one feels acceleration away from that planet and so time away from that planet has to run faster. So again, after her return from vacation on a more massive than the earth planet the traveling twin would found her twin older too even ignoring all the acceleration effects arising from the change of direction of movement.
So once we know how acceleration influences time rate, and we know that velocity is there only as a purely abstract mathematical thing (an integral of that acceleration) the twin paradox disappears. It disappears because one twin really accelerates (her accelerometer shows something rather large) and the other, who stays on the earth, does not (her accelerometer shows about zero). Now we know the difference between the twins' movementsand so we know which one will be younger when they meet: the accelerating one.
To see how this acceleration causes gravitational force let notice that we have here acceleration causing a difference in the time rates along direction of acceleration. There is a strict relation between the two that may be expressed by a simple formula involving only rate of time, acceleration, distance, and the speed of light. Einstein thought that if acceleration is causing a change in the time rate then perhaps the change in time rate would cause an acceleration (and the unavoidable inertial force connected with it). It has been called the principle of equivalence (of acceleration and gravitational force) and it turned out that the nature really works that way.
The presence of mass (or energy, to which this mass is equivalent by famous Einstein's formula E = mc2) causes that time runs slower in the vicinity of that mass, as verified independently by very accurate clocks that became available only recently. It is all that is needed to create acceleration detected by accelerometers and cause inertial forces proportional to that acceleration. It is all without any movement of anything. This way the origin of gravitational force has been explained by the mass of the earth causing the time running slower in its vicinity causing all the conditions that show up when something accelerates. Therefore, on Einstein's principle of equivalence it causes the acceleration, which in turn causes inertial force.
This slowing of time in vicinity of a mass is called gravitational time dilation. This time dilation causing inertial force is what people used to take for the old attractive gravitational force. It may still be called gravitational but there is no way of making it attractive any more since it disappears immediately after the "attracted" object loses physical contact with the "attracting" one. So it does not act at the distance as an attractive force would and as it was imagined that it does.
There is certain popular misconception, even among physicists who are not too familiar with Einstein's theory so it is worth of explaining.
This misconception is that a hypothetical "gravitational attractive force" causes gravitational time dilation and all other relativistic gravitational effects (like e.g. curvature of space). Obviously it can't be so since time dilation by itself causes acceleration and so it causes the inertial force that is exactly equal to that hypothetical "gravitational attraction". If there existed also this hypothetical "gravitational attractive force" then the gravitational force that we observe on the earth would be twice as big as it is: one part caused by the "gravitational attractive force" and one by the time dilation. We observe only one of those parts and since time dilation is verifiable fact (trivial truth verifiable by precise clocks) the other part (the "gravitational attractive force") must be a fiction.
This is how the gravitational attractive force disappeared from physics. That it still stays in minds of some physicists is kind of mystery, which can't be explained by asking those physicists questions since as it was mentioned they claim the lack of time for answering them.
In any case this gravitational time dilation effect (sometimes called time curvature) is the first half of the Einsteinian gravity. It explains fully the same things that Newtonian gravity managed to explain and gives exactly the same numerical results.
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