Introduction to Einstein's General Theory of Relativity

You might emerge somewhere else in space. Somewhen else in time.
— Carl Sagan


The idea of spacetime came from one of Einstein’s teachers Herman Minkowski. At first, Einstein dismissed his idea but later on realized that spacetime would be the perfect model for his theory of gravity. In special relativity, space and time can stretch and contract depending on the relative velocity of an observer’s reference frame; however, special relativity says nothing about curved space and time. To describe curved space and time we need to use Einstein’s general theory of relativity. This theory says that the presence of energy (which is equivalent to mass) curves spacetime and that gravity is the effect of curved spacetime. Objects move in the presence of energy due to curved spacetime—this effect on the object is called gravity. The objects path through spacetime (called a worldline) are governed by the geodesic equation; this describes the kinematics of an object moving through spacetime. The Einstein field equations (EFEs) fully determine what the geodesic equation looks like based off the distribution of energy present. The EFEs, together with the geodesic equations, are called relativistic mechanics. As an analogy, classical mechanics and quantum mechanics are also expressed by a field equation and an equation of motion (Newton’s Second Law and Schrödinger’s equation, respectively). Special relativity is a special case of the more general theory (namely, general relativity).

Minkowski spacetime is a particular kind of spacetime (namely one which is "flat") which is the topic of study in special relativity; but one of Einstein's great realizations is that spacetime can actually curve and be dynamical in the presence of energy and momentum. We’ll be interested in expressing the laws of physics in terms of physical quantities which do not depend on the reference frame (just like for special relativity) and only depend on the geometry and curvature of spacetime. These quantities are members of a class of objects known as tensors. A tensor \(\textbf{T}\) is an object which is the same in all coordinate systems (which is to say all reference frames) because it is defined in terms of geometrical quantities which do not depend on coordinate systems—they depend upon only the geometry and curvature of the space which they live in. For example, a vector \(\textbf{V}\) is a tensor that can be defined entirely in terms of geometrical quantities (namely length and angle) without “imposing” any coordinate system on the space. Suppose, for simplicity, the space is a flat plain; such a space is called \(𝔼^2\). Next, we need two points connected by a line segment with an arrow attached to it. The magnitude of the vector \(\textbf{V}\) is just its length which can be simply thought of as the line segment itself. By drawing a second line through the point at the “tail” of \(\textbf{V}\), we can define the angle \(𝛼\) which gives a sense of direction to \(\textbf{V}\). We could have defined \(\textbf{V}\) using only geometry thousands of years ago before coordinate systems were even invented if we wanted to. (If we want to, we can even think of the direction of \(\textbf{V}\) as just being the arrow that is attached to the line segment.) Another example of a tensor is a scalar \(𝛷\) (which is a tensor of rank 0) which is a value associated with a point in space. For example, the temperature at a point is a scalar. Imagine placing down various coordinate systems where \(\textbf{V}\) is; clearly, the vector \(\textbf{V}\) is the same regardless of which coordinate systems we use. Another example of a tensor is the dot product between any two arbitrary vectors \(\textbf{V}⋅\textbf{W}\) (this is an example of a tensor of rank 2). This is also a tensor because it is the same in all coordinate systems. The reason why the tensor \(\textbf{T}=\textbf{V}⋅\textbf{W}\) is the same in all coordinate systems is because it can be defined entirely in terms of geometrical quantities (namely length and angle). Using the definition of a dot product, we can write \(\textbf{V}⋅\textbf{W} = |\textbf{V}||\textbf{W}|cos𝛼\). Because the length of a vector and the angle between two vectors can be defined using just geometry, those quantities do not depend on which coordinate system is used. Since \(\textbf{T}=\textbf{V}⋅\textbf{W}\) is defined in terms of quantities which don’t depend on any coordinate system, \(\textbf{T}\) is the same in all coordinate systems\(^1\).

To write the laws of physics in terms of these tensor quantities, we need to be able to do calculus on them—that is, take derivatives, integrals, and so on. The branch of mathematics called tensor calculus is the study of calculus on tensors. Another branch of mathematics called differential geometry uses tensor calculus to study the geometry of curved spaces—that is to say, the angles between lines, the length of lines, and so on. Here’s the catch. In special relativity we were studying the physics in flat spacetime—none of the space or time dimensions curved. Since the spacetime is flat, you could take tensor quantities such as the metric or four-vectors, and then imagine “moving them around” through spacetime keeping their length and direction fixed, and these quantities won’t change. This is because there is no curvature and because the geometry does not change. But in the case of a curved spacetime, since tensor quantities (in general) depend on length and angle, and since length and angle can actually change from point to point due to curvature and the changing geometry, it follows that in general tensor quantities will vary from point to point in curved spacetime. The operator which captures the rate of change of a tensor as you “move it around” through each point in spacetime is called the covariant derivative; this operator represents the rate of change of a tensor due to the geometry and curvature changing from point to point in a way that does not depend on coordinate systems of reference frames.

Predictions of general relativity

The physicist Sean Carrol described general relativity as the most mathematically elegant and beautiful mathematical framework ever written about the nature of physical reality. Mathematically, general relativity is indeed very aesthetically pleasing. But perhaps what is even more profound about general relativity are its predictions about physical reality. As mind-bending as the predictions of special relativity are, the predictions of general relativity are perhaps even profounder. Every assemblage of matter has an associated swartzchild radius—just imagine drawing an imaginary sphere whose radius (the swartzchild radius) is determined by the amount of energy (which is equivalent to mass, a property that all matter has) of the assemblage of matter. If you squish all of this matter together so compactly that it fits into this imaginary sphere, then the object will turn into something called a black hole. General relativity predicts that, astonishingly, all of the matter will continue to collapse in on itself until it collapse into an infinitesimally small point-mass of zero size. The mass is finite but, since the volume in which the mass is contained is infinitesimally small, the density is infinite. We shall prove later in these lectures on general relativity that if an observer in one inertial reference frame (say an astronaut in a spaceship) is very far away from a black hole and watches another observer in another inertial reference frame (say an unfortunate astronaut) falling into a black hole, as the falling astronaut approaches what is called the event horizon (a region of space which is an imaginary sphere surrounding the point mass of the black hole and whose radius is that of the swartzchild radius) the far away astronaut will see the falling astronaut’s clocks “freeze in time”—the march of time will have appeared to stop. Also, the astronaut will not only be frozen in time and space but he’ll also be completely flat across the event horizon. But relative to the falling astronaut, they’ll pass right through the event horizon without seeing their clocks slow down or their rulers contract. This is an extreme example of the principle that space and time are not invariant and depend on which reference frame we’re talking about.

General relativity also predicts that objects such as binary star systems (among other physical systems) emit something called gravitational waves—ripples through the fabric of spacetime. Unlike mechanical waves which involve oscillations in material particles or electromagnetic waves which are oscillations in magnetic and electric fields, these waves are oscillations of space and time. As a gravitational wave passes through a material object the space and time container in which it resides will oscillate; the object will contract, expand, contract, etc. and the clocks will dilate, speed up, dilate, etc. The existence of gravitational waves was confirmed by LIGO in 2016.

In my cosmology lectures I also describe how the application of general relativity to all the masses in the universe implies, as Lemaitre once argued, that the galaxies are rushing away from one another due to the expansion of space. Since the laws of relativistic mechanics are time-reversible we can use general relativity to run the clocks backwards to see what the very young universe looked like. According to general relativity, a long time ago when the universe was very young all of the galaxies were on top of each other and the density of matter throughout the universe was so great that the entire universe became hotter than the Sun. We cannot run the clocks back in time too far when the universe was extremely miniscule in size where the effects of quantum mechanics become important since general relativity does not apply to situations where quantum phenomena is apparent.

General relativity also predicts that, by creating the right energy distribution, you could construct something called an Einstein-Rosen Bridge (more colloquially known as a wormhole). This would allow one to connect two separate regions of spacetime. If you could go through a wormhole you could travel to the opposite side of the universe in just a few minutes. If you went through a wormhole not only would you emerge somewhere else in space but, as Carl Sagan once said, you would emerge somewhen else in time. You could in fact travel backwards in time. According to general relativity, this is not the only way to achieve faster than light speed travel. Researchers, using Einstein’s field equations, have come up with ways to build starships which travel faster than the speed of light. One of these designs is called warp drive. The main idea is that the starship could ride along ripples and waves of spacetime. According to special relativity no object can travel through space faster than the speed of light. But general relativity places no limit on how fast the fabric of spacetime itself can move. These spacetime ripples could propagate through the universe faster than light speed, carrying the starship along with it for the ride. This is analogous to how very distant regions of space (more than 10 billion lightyears away) are expanding and “moving” away from us faster than lightspeed, carrying the galaxies along with it. The galaxies motion through space is very small (much less than the speed of light); their supraluminary speeds are obtained by riding along the expanding space itself.

Colonizing the heavens


Stephen Hawking and many other physicists take these ideas very seriously since General Relativity has been proven to be correct (in every experiment done so far at least). The Kardashev scale ranks technological civilizations based on their quantity of power consumption as Type 0, Type 1, Type 2, and Type 3 civilizations. This system was invented by an astronomer named Nikolai Kardashev in the 70s and later refined by Carl Sagan. How technologically advanced a civilization is will also be measured based on their information consumption and the minimization of entropy produced by industrial and economic processes. Jacque Fresco’s vision would represent a Type 1 civilization. If we were able to build a Dyson Swarm around the Sun which a physicist named Freeman Dyson wrote a paper on, this would represent a transition to a Type 2 civilization. Stephen Hawking, who is working with Mark Zuckerberg and Yuri Milner, plan to send a tiny robotic spacecraft to an Earth-like planet named Proxima B in the Proxima Centauri star system. One strategy which has been proposed is to deploy a swarm of thousands of small, self-replicating, nano-ships (called Von Neumann probes). It is anticipated that by the 2050s, there will be trillions of such probes dispersed throughout one of our solar system’s asteroid belts. This strategy of sending a myriad of such probes from world to world is, according to Michio Kaku, the most likely way humanity would transition to a Type 3 civilization and spread throughout our galaxy. These phone-sized nano-ships would have miniaturized 3D-printers which can excavate resources from other worlds and use them to 3D-print factories which are capable of making more nano-ships, which in turn can 3D print more factories, and so on. Assuming that these probes traveled at speeds in between light speed and 10% of light speed, we would eventually colonize the Milky Way galaxy in about 100,000 to one million years. We could at long last, in locution of Carl Sagan, “sail across the starry archipelagos of the vast Milky Way galaxy.” But the cosmic speed limit—that no material object can travel across space faster than light—means that it’ll take very long to colonize our galaxy. But the key to breaking the cosmic speed limit barrier and achieving FTL speeds likely resides in Einstein’s general relativity. Creating a wormhole or warp drive would allow us to colonize the Milky Way much faster.

This article is licensed under a CC BY-NC-SA 4.0 license.

Sources: PBS Spacetime, Futurism


1. If you are interested in learning more about "calculus on invariants," or tensor calculus, I highly recommend watching this lesson and subsequent lessons from this lecturer. I personally learned most of what I know about tensor calculus from him.