Time: The reason behind the Gravitational Force

General Relativity can be said to be one of the greatest theories. The interplay of space and time that gives rise to the fundamental force of the universe is geometrically the most elegant theory out there. It is intricately complicated but beautiful in the sense that the underlying principle behind the theory is simple and magnificent. But the math of the theory is so complex that in 1919 when someone asked Sir Arthur Eddington whether it was true that only three people in the world understood the theory of general relativity, he allegedly replied: ‘Who’s the third?” This theory is, thus, heavily misunderstood, with almost all the analogies offered in order to understand this theory mistaken in some or the other way.

The Flaw in the View

The very first illustration someone shows when explaining gravity in the sense of general relativity is that of a massive body placed on a stretched rubber sheet.

The massive body causes a depression in the rubber sheet, and this dip causes other bodies to move towards it. But in reality, this view is heavily flawed just like any other illustrations.
First of all the two-dimensional sheet curves downwards, i.e. into a third dimension. So for this view to be true, there must be a fifth dimension, apart from the four spacetime dimensions, into which the spacetime must curve which is not the case in reality, where the four dimensions of spacetime curve within themselves and not into some other dimension.

The sheet here curves only because an external force (here Earth’s gravity) pulls down the mass. In a vacuum, the sheet would not have curved at all. This view implies that there must be an external gravity-like force that pulls the mass into some other dimension causing the spacetime to curve, which again is entirely a misinterpretation. 

This curving in two dimensions of space and zero dimensions of time actually looks like this, 

which is completely different from the rubber sheet analogy. Notice that there is no dip or external force acting in the downwards direction, but only a curve in the spacetime grid. A body moving with some velocity, as it gets close to the gravitating body, follows a straight path (denoted by the space-grids) in curved space. This looks to us as if the object is attracted to the gravitating mass. 

It’s all fine up to now, but here we have a problem. A body, initially at rest at some distance r from the gravitating object, has no reason to get attracted to it. It is fine sitting just there, even if the spacetime is curved, i.e. there should be no force on a stationary object. This should mean that the force of gravity should have been a velocity-dependent force, which is not quite the case in reality.

So where are we flawed?

The answer is simple. 

Time. 

We ignored time as a dimension and did not consider its curvature at all. Gravity without time is incomplete. General relativity without time is incompatible. Gravity can be only explained with a curvature in “spacetime” and not “space” only. How?

Consider a lift in empty space. Einstein’s special relativity states that there is no way a person inside that lift could know if he were moving or not. But what happens if the lift was to be accelerated? The equivalence principle states that an accelerated movement is equivalent to gravity, so the person inside the accelerated lift still has no way to find out if he is moving or is gravity acting on the lift, thus ensuring that the condition is satisfied.

What if we set up things in a way to violate the condition? Could we enable the person inside the lift to tell if it is gravity or acceleration he is experiencing? Consider a photon emitter placed on the top of the lift and a detector at the bottom. Both are equipped with clocks that measure the time interval between emitting of the two photons and between the detection of two photons. Since the lift is accelerating, its velocity keeps on increasing and hence the time interval between the emitting of two photons and the detection of the same two photons is not the same.

If there was a one-second gap between the two photons when emitted since the second photon has to travel a lesser distance than the first photon, the interval between the detection of the two photons is lesser than one second. To the person inside the lift, it would look as if the clock on the bottom is running slower than the clock on the top. To prevent the person from knowing if the lift was accelerated or if it was gravity, the same thing should happen in a gravitation field. I.e Time should run slower the closer you are to a gravitating body. This exactly is what is called gravitational time dilation, and it is an inevitable consequence of the theory. 

Thus a massive body not only warps space but also sets up a time gradient, where time moves slower the closer to the body. 

And this time gradient is exactly what causes the force of gravity. 

How is this Time Gradient responsible for the Force of Gravity?

Imagine a tea-cup floating freely in empty space, with two clocks attached – one on the top of the cup and one on the bottom. There is no gravitational field and so both the clocks are synchronised and ticking at the same rate, i.e both clocks are moving with the same speed through the time dimension. Introduce a gravitating body, it sets up a time gradient where time flows slower at points closer to the gravitating body and faster at points away from it. The situation can be better thought of as two boats riding downstream, with both of them connected by a rigid rod. Let the stream be flowing in t direction, and the perpendicular direction is one of the spatial dimensions. The two boats are the two clocks and initially, the boats are entirely moving along the stream, i.e. its four-velocity (velocity in the 4D spacetime) is entirely in the time direction and its spatial velocity is zero. Now if there was to be a velocity gradient in the stream, i.e. if the boat on the right were to be going faster than the one on the left, the rigid rod connecting them both would cause a torque to be exerted, turning both the boats to the left,

i.e the motion which was initially only time-like has now obtained a spatial velocity just due to the fact that there is a velocity gradient in the time direction. 

The same is the situation with the tea-cup with the clocks. When in a gravitational field, the clock on the top moves faster that the clock on the bottom. Since the body is rigid, this causes the body to gain velocity in the spatial direction, and this is always towards the direction of a slower time. Every body can be considered as made up of many clocks, every subatomic particle is a clock in itself as it feels the flow of time. So every composite body under the time gradient experiences this force of gravity. The situation can be generalised even to point particles since these can be considered as two clocks separated by infinitesimal distance.

Thus when someone says that gravity is the curvature of spacetime, it implies that not only the curvature of space causes is important, but the time gradient set up by the Energy source also plays a prominent part in giving you a huge spatial velocity for a small price in temporal (time) velocity, the slowing down of time. Gravity is the force that arises as a result of a body’s tendency to move towards regions with slower time in the presence of a time gradient.