A Sad Introduction to General Relativity
Sir Isaac Newton’s theory of gravity reigned supreme for over 200 years before its faults were finally exposed. Scientists had exhausted all their creativity, to come up with experiments to test Newton’s laws in surprising scenarios. Newton’s laws were effectively used to calculate the 2 orbits of planets, comets and moons with breathtaking accuracy and precision. He was able to explain all of Kepler’s laws of planetary motion with his simple laws. Even today, Newton’s law of gravitation is all you need (in addition to A LOT of money) to launch spaceships and land rovers on other worlds. So what went wrong? Why did Einstein have to come up with a radically new theory of gravity? What are its implications? How was it tested? Is it the end or is something more to come? And the most important question of all, why on Earth does space-time have to be curved?
Let’s start at the very beginning. The first time when people suspected that Newton’s law of gravitation was flawed was when an anomaly was detected in the orbit of Uranus. The anomaly could be explained with 2 explanations: either Newton’s laws had to be tweaked slightly or there was an undiscovered planet with a specific mass and a specific distance beyond Uranus. Putting their faith completely in Newton, a few mathematicians (independently) calculated the orbit of this planet and wrote a letter to their local observatory saying: “Point your telescopes at this point in space at this time, and you’ll find a planet”. And lo and behold Neptune was discovered! Victory for Newton!
Then, a few years later a man named Urbain Le Verrier (the same mathematician who “discovered” Neptune) described a peculiarity in the orbit of Mercury. Le Verrier found that the perihelion (the point in the orbit of a planet at which it is closest to the sun) of Mercury was changing, in a way that could not be explained by Newton’s Laws. Ah, history repeats itself (or so they thought). People assumed that the anomaly in the orbit of Mercury was caused by an undiscovered planet, which they called Vulcan. And so the hunt for Vulcan began and ended with excitement. The hunt ended, not because the planet was found, but because it WAS NOT found. Newton’s Laws had finally reached their breaking point, and it was finally known that the Universal Law of Gravitation wasn’t so universal after all. Another problem with Newton’s Laws is that they assume that the speed of light was infinite and time was absolute. These ideas were later dismantled by Einstein in his Special Theory of Relativity and were simply wrong ideas that Newton had accepted. We, the inhabitants of the future with the power of hindsight must remember, however, not to be harsh on Newton. He reasoned his way to a mathematical formulation of gravity that works well in most aspects of life and united the notions of falling apples and orbiting planets with one simple equation. All this, in addition to him creating a design of the telescope that’s still in use today, revolutionising optics, casually inventing calculus and developing the laws of motion. Oh and by the way, after doing all that he turned 26.
Before diving into general relativity, we must explore the concept of an inertial frame of reference. An inertial frame of reference is a fancy phrase which simply means a non-accelerating frame of reference. Newton’s Second Law of Motion, F=ma, is only valid when the observer is in an inertial frame. Let’s think about this with an example:
Imagine yourself on a train. If the train begins to undergo uniform acceleration, relative to the train’s interior you will accelerate backward, even though you can’t identify any horizontal forces on you. So inside the train, F is not equal to ma. From the perspective of an observer on the ground, you don’t accelerate at all. Instead, the train car accelerates forward to meet you, and hence F=ma.
All acceleration is measured with respect to inertial frames, and by definition, inertial frames of references do not accelerate with respect to one another. There’s a quick and dirty test that you can do to check if your reference frame is inertial or not: pick up an object, and let go of it. If it stays in the place where you left it, your reference frame is inertial.
Now that we know about the limitations of Newtonian gravity and inertial frames, let us take a look at a debate that transcends time itself, a debate between Newton and Einstein, a fight for the validity of inertial frames of references.
Einstein: Newton, if we put a cup of coffee, a piano and a jaguar in a train which begins to accelerate forward, from the perspective of someone within the train, everything would accelerate backward at the same rate. And what other phenomenon are we aware of that makes things accelerate at the same rate? Yes, exactly! Gravity makes everything (at the same height) accelerate uniformly. The forward acceleration of the train is identical to a gravitational field that points backward. If we consider the acceleration due to the train’s motion, a ‘fake’ gravitational field and add it to Earth’s ‘real’ gravitational field, we get a net gravitational field that points diagonally downward. In fact, if we attach a helium balloon to the floor of a train undergoing uniform acceleration, we see that the balloon aligns itself in the direction provided by the vector sum of the accelerations of the ‘real’ and ‘fake’ gravitational fields. I thus propose that the acceleration due to Earth’s gravitational field is also ‘fake’, a side effect caused by the Earth’s acceleration upwards.
Newton: You’re perfectly right in your analysis Einstein, however, I must remind you to bring to your attention a few things. First of all, thinking of the acceleration directed backwards due to the train’s motion forward as a gravitational field is nothing more than a calculation trick, you must not lay the foundations of your bold claims with such hacks. Second of all, inertial frames are the standard for measuring true acceleration. So unless you can provide me with an inertial frame relative to which the Earth is accelerating upwards, it simply isn’t.
Einstein: Not so fast, Newton. How about a reference frame that’s in free fall? Now hear me out. If I put you and a few objects in a box and throw you off a cliff, you would see everything in the box just float. For all you know, you’re in the middle of intergalactic space floating for no reason. This frame of reference even passes the quick and dirty test that YOU set up for an inertial reference frame. I can confidently say that the falling frame of the box behaves like a stationary inertial frame. And relative to this inertial reference frame, the Earth is accelerating upwards. So there you have it, an inertial frame relative to which the Earth is accelerating upwards.
Newton: I’d congratulate you, Einstein, if there weren’t two fatal flaws in your argument. First of all, down isn’t down, it’s radially inward. Two objects in free fall are falling toward the Earth on two non-parallel radial lines, directed inward towards the Earth. So from the perspective of an observer within the box, the distance between two objects inside the box will not remain constant, they will accelerate toward each other. Second of all, by your criteria, orbiting frames of reference should also be considered inertial. But an orbiting reference frame accelerates relative to a reference frame in free fall, but inertial frames cannot, by definition, accelerate relative to each other.
Einstein: Well, huh…….that’s a good point.
So, it looks like Einstein’s viewpoint was shot down by Newton with beautiful reasoning and logic, doesn’t it? Well, not so fast. Einstein was a very persistent man. He hated seeing two identical phenomena (over here ‘regular’ acceleration and acceleration due to gravity) given two very different explanations. His patience (for 7 years!) with the question led him to a counter-argument. He realised that the rule that inertial reference frames can’t accelerate relative to one another is only applicable if the world has a ‘flat’ geometry. If, however, the world is a non-Euclidian AND curved spacetime, then inertial reference frames can basically do whatever they want. In a non-Euclidian and curved spacetime, our intuitions about “basic” and “obvious” things like straight and parallel lines break down. This was the loophole that Einstein found, he assumed that the world has a non-Euclidean geometry and built his theory up from there.
If you accept Einstein’s picture of reality, gravity is really not a force but emerges from geometry. Just like the jolt that you feel in a uniformly accelerating train is not due to a force, acceleration due to gravity is not due to a force but due to you, dear reader, inhabiting an accelerating frame of reference (assuming of course that you are not in free fall or in space as you read this article).
This theory, that has come to be known as general relativity, paints a beautiful picture of the world. It’s passed every single test that’s been thrown at it, and most of its predictions about the world, have been tested and verified. Solving the equations of general relativity alone, physicists were able to predict the existence of black holes, gravitational waves, and even correct their calculations for the orbit of Mercury. Einstein’s worldview has been confirmed several times over, the most recent confirmation being the detection of gravitational waves.
Our theory about gravity is far from complete. General relativity is still a classical theory and completely ignores the uncertainty principle of quantum mechanics. We still do not know what gravity at the quantum scale would look like, but there’s good work in that direction. Years of sweat and desperation lead physicists to the theory of quantum gravity that reigns supreme today: string theory. When physicists take on the colossal task of solving the maths of string theory, the equations of general relativity pop out almost like magic (if they didn’t people would’ve just added a couple of extra dimensions to the maths until they did!). String theory, however, is far from complete and is still in its infancy.
After playing with Einstein’s equations, physicists were able to discover real-world objects and phenomena and were not experimentally confirmed for decades after their theoretical birth. Einstein united geometry with gravity, showed how gravity is nothing but an illusion and blew the collective mind of all of humanity with his groundbreaking theory that continues to seduce us with its simplicity, beauty and elegance.
Originally published at http://shortdotcircuit.wordpress.com on October 27, 2020.
