A soft intro to General Relativity

Source: Wikipedia

We are pretty much obsessed with seeing things from a Newtonian point of view that it has become a natural intuition. Newtonian view of the world seems obvious and satisfying. However, reality can be different than what it seems. For example, a body need not necessarily require a force to change its state from rest to motion. “The Theory of General Relativity” (GR) is a clearer viewpoint to seeing things around us and this overthrows even the most fundamental principles of Newtonian physics.

Albert Einstein is a celebrity physicist and a very well-known name. He started his work on GR because of the discomfort he had in accepting the Newtonian view of gravity and space. And the theory he put forward is arguably the greatest physical theory ever discovered, often considered as the greatest insight to ever cross a human mind. The scope of GR is huge in the scientific world ranging from understanding time, space, and motion to black holes, gravitational waves, and the big bang.

The masterpiece is known as Einstein’s Field Equation and it looks like this:

GR has been finely scrutinized by a number of experiments over these 100 years and it has been able to stand out on every occasion.

What’s here?

This article is intended for absolute beginners. We are not going to discuss the big names as black holes or gravitational waves or the big bang. We are merely discussing how general relativity interprets the daily world around us, mainly the two things: the falling of objects and our ‘feeling of weight’.

The real essence and beauty of GR lie in its overly complicated Einstein field equations. But I will refrain from writing any mathematical equation here onwards. I will try to break down the concepts into simple analogies and explain them with proper visualization.

Gravity is not a force

Newton described gravity as a force that attracts masses towards each other. Einstein refused to accept this very fundamental postulate of Newtonian gravity. What Einstein believed was:

This, despite looking simple, is a bold statement and it gives rise to a lot of questions, the very first one being “If not a force, what makes things fall?” We will come to that later. First, let us turn our attention to the curvature in space-time. There is a famous rubber sheet analogy that is often used to explain the concept.

Source: Wikipedia

To explain gravity from a GR perspective, we need to extend three dimensions of space with an additional dimension of time. Space-time refers to this unification. This can be difficult to visualize at the moment but bear with me.

In Newtonian mechanics, space was considered as a platform where things happen. But, there were no happenings within space itself, i.e. the space was static and undisturbed. However, GR explains space-time as a dynamic entity that changes with the presence of mass. It is like being a rubber sheet that is usually flat. But in presence of heavy mass, the sheet curves as shown in the picture above. This curvature signifies the presence of the gravitational field and is responsible for all the gravitational effects we observe. An example is in the video linked below in which a test mass projected in space-time tends to follow a curve and orbit the huge mass. Actually, the mass tends to follow a straight line, but the space is so curved in the region that it ends up moving circularly. Please note that the figure is just a visualization and does not depict the actual picture. Here is a short video to illustrate the curvature in space-time.

https://www.youtube.com/watch?v=-m3SddsTSAY

If not a force, what makes things fall?

It is well-established according to Newton’s laws of motion that falling of a body is a result of a force acting on it. Thus, we have a contradiction between laws of motion and GR. The fact is that laws of motion as described by Newton are no longer accepted in GR. Previously, we overthrew Newton’s gravitational theory and now we are overthrowing the laws of motion as Newton described*.

As per GR, falling of an object is its natural motion in space-time and no external agent is required to make it happen. Let’s imagine a body lying still in gravity-free space which has no spatial motion but obviously, is moving forward in time. Do we ever question what agent or force is involved in pushing it forward through time? - No, because we know time flows naturally without requiring an external agent. This is a similar scenario: in presence of a gravitational field, spatial motion also happens in addition to temporal motion without the need for an agent. The nature of motion a body possesses is determined by the nature of space-time around it. In the gravity-free region, space-time is flat and hence only temporal motion. In presence of mass/gravity, space-time is curved and as a result, we have spatial (falling) motion as well. What we can also infer from here is that space and time dimensions are not very different and hence moving forward in time is similar to moving in space.

To clarify this further, an inclined plane in Figure 1 below would serve as a good example.

Figure 1: Inclined plane

The ball cannot be taken vertically down to the ground. This is because the structure of the plane is such that when we move it downwards, the ball will automatically acquire some forward movement as well. In other words, you cannot bring it down without moving it forward. Similar is the case with the motion of objects in gravitational field. The radial space dimension and time have such an interwoven structure that you simply cannot move forward in time without moving radially downwards. That means any object if left undisturbed, must fall radially as time passes. This is a mind-boggling insight of GR. It can make a little more sense now regarding why space and time are unified together as a 4-dimensional structure: because they are so related that one has an effect on the other. However, it is still difficult to visualize how time and radial space dimensions are so interlinked together. This is because of our sensory limitations to realize time as a dimension.

  • The essence of Newton’s laws of motion is not completely overthrown in GR. But they are not valid as Newton described. While Newton’s laws only accounted for motion in spatial dimensions, the laws should be extended to account for motion in temporal dimension as well to comply with GR.

The Equivalence Principle

In the beginning stage of the development of GR, Einstein had a thought which he later described as the happiest thought of his life.

With this seemingly simple insight, he was able to take a great leap in understanding the nature of gravity, the major realization being that falling freely “nullifies” gravitation.

Figure 2: Equivalence Principle

The upper half of Figure 2 depicts two scenarios: on the left, we have an elevator with upward acceleration g in gravity-free space. On the right, there is another elevator lying still on the earth’s surface. If we see things from the elevator’s frame of reference, we notice the same things in both of the cases: if there is a ball inside the elevator, it will fall down with acceleration g. It basically means that when we are inside an elevator and see objects falling at g, there is no way we can know if we are accelerating upward in gravity-free space or if we are actually inside Earth’s gravitational field. We can even weigh ourselves inside the elevator and the weighing machine will have the same reading in both cases.

Now in a different setting, suppose we are inside an elevator and see objects lying still as we see in the lower half of the figure. In this setting, there is no way we can tell if we are lying still in gravity-free space or if we are freely falling inside a gravitational field.

From both of these scenarios, we can infer that the effects of gravity and acceleration are indistinguishable. This is what is called the Principle of Equivalence (PoE) and forms a pillar of General Relativity. GR goes on further to explain that gravity is not a separate phenomenon but a result of acceleration itself.

We need to modify the Newtonian definition of acceleration accordingly to make it consistent with the above statement. Once again, Newton defined acceleration for flat spatial co-ordinates only, but it must be extended to temporal coordinates and defined in curved space-time to comply with GR.

The feeling of ‘weight’

We learned previously that a body if left undisturbed, must fall radially as time passes. Let us now discuss what happens at the surface of the Earth. When a body falls down and strikes the surface of Earth, there is a disturbance in its natural motion and the body starts feeling its weight. According to the principle of equivalence, the point at which the body strikes the ground is the exact point when it transitions from inertial motion to accelerated motion, and this acceleration is the cause of the feeling of weight. Thus, our feeling of weight is actually the result of Earth’s surface accelerating us radially upward just like we feel weight inside an elevator accelerating upwards in gravity-free space.

If every point on Earth’s surface is accelerating radially outward, the earth should be expanding. Why is it not the case?

We are accelerating radially outward, but we are not ‘moving’ radially outward. We can explain the fact once we realize that we need not necessarily move out radially to have radial acceleration. The dimension of time has a beautiful role to play here as well. Let us take an analogy to understand this.

Figure 3: Circular motion

Imagine of a spaceship moving in a circular loop in gravity-free space. And we have a creature named C living inside the spaceship which only has the sense of one spatial dimension, the radial dimension. Let us be more clear about it here because this is the central assumption we are making. Just like we human beings can feel and realize 3 dimensions, the creature is so limited naturally to only realize 1 dimension, the radial dimension in this case. Thus, the creature living inside has no sense of angular direction as shown in Figure 3.

While we consider the spaceship to be having a two-dimensional motion, let us see things from C’s perspective. Everything C can see is the radial dimension, and throughout the motion, it is at a constant radial distance from a certain point. So, C considers itself to be at rest. Note that the motion is only in the angular direction which C has no sense of. From our perspective, we can infer that C must also be experiencing an “outward pull” as a result of circular motion*. For C, yes it is experiencing outward pull but it does not have a sense of circular motion to explain it. So, C will end up concluding falsely that the spaceship is pulling it outwards with a force.

*Circular motion is a good example where a body accelerates radially but it does not move in radially.

If the above example didn’t click you already, let me tell you that the conclusion C ended up with is the same conclusion Newton had reached for explaining this “pull” towards the Earth. Let’s compare ourselves with C’s condition. Just like C experienced, we are at a constant position in space and experiencing a pulling force. More importantly, C had completely failed to realize about its circular motion (i.e. motion in angular direction) because of its sensory limitations, are we missing out on some dimension as well? — Yes, TIME — We are completely missing out on our motion in time because of the same reason. For C it was its motion in the angular dimension that was causing radial acceleration which it misunderstood for a force. For us, it is our movement in time that is causing radial acceleration. And it took the genius of Einstein to clear out the misunderstanding despite our sensory limitations on realizing the dimension of time. This further shows how close time and space are linked. This analogy is for a simple visualization purpose and does not depict exactly what happens on Earth’s surface. Nonetheless, the analogy serves to have an insightful understanding.

General Relativity is a difficult concept to grasp because it goes beyond the things that we generally see and feel. GR has taken a great leap ahead in understanding space and more importantly, understanding time.

Learning GR can be onerous at beginning. The complicated mathematics are very entangling that we may fail to see the bigger picture. That was the reason behind the motivation to write this article. And as intended, the article tries to provide a math-free visualization of the fundamentals of GR.

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