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### From Special Relativity to General Relativity

Einstein proposed his Special Theory of Relativity (SR from now on) in 1905 where he gave new set of equations for mass, time and velocity for objects moving with constant speed.
The concept of spacetime is an important aspect of SR. It says that space and time are related. Two interpretations of spacetime are useful to remember.
1. Each one of us carry our own coordinates and our own time. There is no absolute space or absolute time.
2. My time is a mixture of your spacetime, your spacetime is a mixture of my space and vice-versa.
Brief summary of General Relativity (GR): I will give a brief idea of GR, before going to narrate the story of its evolution from SR.

GR in brief says that bodies with mass warp the spacetime around them. The gravitational force on a second body is nothing but the motion of second body in the warped spacetime of the the first body.
It’s commonly explained with the example of a dense ball placed on a trampoline sheet. The ball creates a dent on the sheet. A small ball in the surrounding falls into the dent. Gravity acts in a similar fashion.

Just a clarification — In case of trampoline, the gravity from beneath the sheet makes small balls fall into the dent. In case of spacetime, you don’t need a force from beneath. Warped spacetime means that the first body changes the shape of the spacetime in which the second body moves. So, the second body has to move along the curved spacetime, which appears to be gravity.

In other words, the gravity of first body acts on the second body by shaping the trajectory of the second body. It is similar to shaping the curves of a railway track. By changing the shape of tracks, you change the trajectory in which the train moves. This is equivalent to applying force on the body to change its trajectory.

One might argue, let someone else change the shape of rail tracks. It can change trajectory only once the train starts moving. How can changing trajectory make the train move in first place?

Note the difference between space (x,y,z) and spacetime (x,y,z,t) we discussed in previous post. We are accustomed to thinking in space framework. So, we think that we are at rest. In spacetime framework (x,y,z,t) you are always moving because time is moving. Therefore, you have no option but to start moving along the trajectory and no option but to travel along the warped trajectories, caused by gravity.

The following post is structured as follows
1. Why did Einstein feel the need to come up with GR?
2. The two important principles for deriving GR — equivalence of acceleration and gravity, and; link between space and time.
3. Happiest thought of my life: Deriving equivalence between acceleration and gravity.
4. Proving that space is warped.
5. Proving that time is warped.
6. Understanding gravity using an example.
7. Einstein’s equations for GR
8. Experimental evidence for Einstein’s GR.
9. Applications of GR
10. Conclusion

#### I. Why did Einstein feel the need to come up with General Relativity (GR)?

In SR, Einstein revised all formulae of space and time in kinematics, by including the time element in them. But then he realised that the original Newton’s equation for gravitation force F = GMm/r² doesn’t have the time element in it. He was uncomfortable with this.

The significance of “no time element” in Newton’s equation for gravitation means that any shift in motion of Sun would be instantaneously felt by earth. But as per Einstein, nothing can travel with speed greater than that of light. How can Earth instantaneously feel the gravitational force of Sun?
In SR, Einstein updated all equations of Kinematics (motion of bodies with constant speed), including the time element. Now, he set himself up to replace all equations of Dynamics (motion of accelerating bodies) to include the time element.

#### II. Two important principles for deriving GR

Two principles are crucial to derive GR from SR.
1. Equivalence of acceleration and gravitation: It means that effects observed in accelerated reference frames are applicable in same fashion to reference frames with gravity and vice-versa.
2. Link between space and time: As per SR, space and time are interlinked. It means that if you prove that space is warped, then it also means that time is warped and vice-versa.
The second flows from SR. The first is discussed below.

#### III. Equivalence of acceleration and gravitation

Equivalence of acceleration and gravitation is an important element of GR. Einstein knew how to update equations of motion in Dynamics with time element but he didn’t know to do that when gravity comes into picture. He needed a way to link acceleration and gravity.

The happiest thought of my life

Einstein described the moment he realised how to link gravity with acceleration, as the happiest thought of his life. The story goes as follows.

A painter fell down from top while painting. Einstein went to meet him. Einstein asked the painter — how did it feel when you were falling? The painter replied — I didn’t feel the motion.

Einstein came back to his room and started thinking on painter’s answer — why did the painter not feel anything? Einstein reasoned that when you are standing on surface, you feel gravity because you are constantly being pushed by the surface. When you are in free fall, there’s nothing to push you, so you don’t feel the gravity.

In other words — if you were to stand on a weighing machine and fall feely along with the weighing machine, the machine wouldn’t show any weight.

It means that one can counter the gravity by getting into motion.

Einstein then reasoned — the reverse must also be true. One can mock up gravity by accelerating the surface beneath the person. For instance, if you are in an elevator in free space and if your elevator is accelerated upwards with g, you would feel as if you are in presence of gravity.

Thus it means that there is an equivalence between gravity and acceleration. What’s applicable in accelerated frame is applicable to gravity and what’s applicable to gravity is applicable to accelerating reference frame. Here comes the link between gravity and acceleration.

Einstein called this moment, the moment he realised the equivalence between acceleration and gravity- as the happiest moment of his life.

#### IV. Space is warped

Consider two disks. One is rotating uniformly and the other is at rest in space.

Divide the border of rotating disk into small unit lengths. If the disk is rotating, each of these unit lengths will contract as per SR, because they are moving.

The motion of the points is in tangential direction. So, the length contraction should be in the the tangential direction. It means that the length along the tangential direction which is the circumference should contract. At the same time, length perpendicular to tangent (along radius) should not contract, because there is no motion along that direction.

In other words, motion of disk contracts its circumference keeping the radius constant!

How is that possible?

It’s not possible in Euclidean space which is plain. It is only possible in bent space.

Imagine that you squeeze the disk in all directions to decrease its circumference and end up decreasing the circumference. Where does the matter in between the disk go? It bends inwards, creating a saddle.
Since the circumference is reduced, the corresponding radius should also be reduced! But no, wait! It happens in plain space, not here. Here you measure the radius, distance between two points on the circumference passing through the centre, by traveling inside the saddle and coming out. So, the corresponding decrease in radius due to shrinking in circumference is compensated by the distance travelled inside the saddle. Thus, the radius remains constant.

Note that the centripetal acceleration of the points on the border is along the radial direction.

It means that acceleration along radial direction, which has caused the circular motion of the body also ends up curving the space in between.

Therefore, accelerating bodies warp the space.

Go back to the equivalence principle of gravity and acceleration. What’s applicable in accelerating frame is equivalent to the effect in other inertial reference frames with gravity equal to the acceleration of the non inertial frame.

Therefore, accelerating bodies warp the space implies that bodies exerting gravitational force also warp the space.

Boom!

Thus, masses warp the space.

V. Gravity warps time

Consider a lift accelerating upwards. Suppose that you shoot light from the bottom of a lift onto its top. Remember that light always travel with constant speed.

Calculations show that distance travelled by the light is greater than twice the height of the lift. It means that clock inside the lift measures more time, that is it becomes slower. Clocks in accelerating frames slow down.

From the principle of equivalence it follows that clocks in influence of gravity slow down.

Gravity warps time.
Clocks tick slower at places with greater gravity.
Alternatively, we can also say that — From the 2nd principle discussed above —the link between space and time , when we proved above that space is warped, it should also mean that time is warped.

We now have the following conclusions.
Gravity warps space.
Gravity warps time.
Gravity hence warps the spacetime.

#### VI. Understanding gravity warping spacetime, using an example.

Let’s say that you drop two bodies parallely on to the ground. As per Newton, they both move towards the centre of the earth and meet there, as shown below. Balls fall along the dotted line. [Ignore the text — moon]
How is that two bodies dropped parallely meet?

Newton says that it’s due to gravity that pulls them towards the centre of the earth.

Einstein says that warped spacetime bends its trajectory making them meet.

Warped spacetime means that two balls are not travelling in a plain Euclidean space. They are travelling in a bent space, similar to travelling on a sphere as shown below. A & B start falling towards the center along the warped spacetime.

VII. Einstein’s equations of General Relativity

This section is only to demonstrate the complexity of Einstein’s equations. They are not meant to be understood. Feel free to skip this, if you don’t like to look at the Math.
The left hand side is geometry (curvature) and right hand side is energy-momentum. This is a compact form of equation. Each of these terms can be further broken down.
(1st step — expanding G (myu-nyu)
(2nd step — expanding R term)
which means
(3rd step — expanding R -myuNyu in 2, which is called Ricci Tensor)
where
[this is called Riemann Curvature Tensor]
Further, the Taus in Riemann curvature Tensor can be expanded as
Probably, now you understand why they are called complex! It can’t be even put in single equation. We had to expand it step-wise. There are 10 parameters. You have to solve 10 equations to get a solution.

#### VIII. Experimental evidence for GR

1. Orbit of Mercury: Orbit of Mercury was slightly off the path calculated using Newtons laws. People thought that probably it’s due to gravity of another mass which we haven’t discovered yet. But Einstein’s equations calculate it precisely.
2. Shift in apparent position of star: Light of stars passing from around Sun bend because Sun’s gravity bends the path in which the light travels. Newton also says that light can bend due to attraction but Einstein’s equations show that it bends due to curved spacetime and they bend twice the amount Newton predicted. Experiments proved that the bending of light is in agreement with Einstein’s equations.
3. Gravitational waves: Einstein’s GR says that rotating bodies in spacetime send out gravitational waves. We recently detected gravitational waves. The waves detected match with the template of waves constructed using Einstein’s equations for two black bodies.
4. Whirlwind in space: When a body rotates in spacetime, it grabs the spacetime around it and twists it, causing the surrounding spacetime to rotate. Gravity Probe B detected such effect.

IX. Applications of GR

GPS is one of the main applications of GR. GPS is able to give us precise locations because it uses GR to correct of time dilations due to motion and gravity. Without GR, its calculations would go haywire.

A satellite in orbit faces two effects on its time — one effect is the Special Relativity effect due to its motion, the second is due to its presence in gravity.

Determining GPS location is all about measuring the time taken by the waves to reach the satellite from the object. So, precision in time is important to determine the location.

SR and GR provide us the necessary corrections that should be made to the calculations, enabling us to precisely determine the physical location.

X. Conclusion

The happy thoughts of a man sitting in a patent office resulted in these theories that are completely counter intuitive and shatter our imagination. But, surprisingly turning out to be true.

GR is now used in several fields. GPS is famous application as discussed above.

GR also predicted black holes, worm holes and so on. People didn’t initially believe it but they ended up finding evidence for black holes later. Black holes are a completely strange phenomenon altogether!

If we think back, all of this sounds surreal.

100 years after Einstein presented his GR (1905), we are still doing his homework!

Images taken from here
Einstein’s equations from Brian Greene’s World Science Festival