General Relativity For Dummies: An Intuitive Introduction

To me, the theory of general relativity is one of the most beautiful theories of nature ever invented, but it is often seen as one of those obscure topics that are not easily accessible to most people.

As a brief introduction, general relativity is the most accurate theory of gravity so far, introduced by Albert Einstein in the early 1900s. General relativity explains gravity as a property of spacetime rather than a force, namely, as the curvature of spacetime, which is caused by matter and energy.

Now, while the mathematical structure of general relativity is, without a doubt, a complicated mess to most beginners, understanding the theory itself and the physics behind it is actually quite simple. This is what we will explore in this article.

From reading this article, you will learn:

• The underlying reasoning behind what general relativity is all about; why we use tensors, how and why gravity has something to do with spacetime curvature as well as what the actual equations of general relativity really tell us in an intuitive sense.
• How general relativity is used to model gravity; we will look at the physical meaning behind all the complicated math, how they relate to gravity and also how the concepts of general relativity relate to the more usual Newtonian theory of gravity.

In a way, this article will work as an in-depth overview of general relativity and more importantly, it will give you the big picture and the important tools needed to learn more of whatever topic you find interesting.

Each subtopic will contain practical examples and in-depth additional sections for those who are interested in exploring the details and applications as well as how the math works.

These will not be necessary to understand the basics, however.

Also, while this article doesn’t necessarily require previous understanding of much of anything, it will help greatly to have some knowledge of special relativity.

For that, I’d recommend reading my introductory article on special relativity.

I also recommend checking out my full Mathematics of General Relativity -course if you’re interested in a dedicated learning experience to take your knowledge of general relativity to a deep level.

Why Is General Relativity Important?

Before we get started on what general relativity actually is, there is an important aspect to be discussed: why should you even care about this topic?

Sure, you’ve clicked on this article, so you’re probably interested to learn more, but still; what makes general relativity an important theory?

Here is a list of a few of the most important applications of general relativity:

• Newtonian gravity is incomplete and sometimes gives incorrect predictions, which general relativity describes correctly: even in our solar system, Newton’s laws are incapable of predicting the orbits of planets precisely (generally, planetary orbits are actually not ellipses).

• General relativity describes new phenomena, which the old Newtonian law of gravity could not predict: examples of this are black holes (in principle, Newtonian gravity can predict they exist, but not describe them to the extent of GR) and gravitational waves (this is a totally unique phenomenon to GR).

• General relativity is at the heart of most modern research in theoretical physics: you’ve probably at some point heard about the whole “quantum gravity” thing, which arguably involves a lot of active research in general relativity. Other huge areas of research in GR are cosmology and astrophysics, which study topics like black holes, dark matter, dark energy, the Big Bang and even the structure of planets and stars.

Arguably, regardless of how practical you view general relativity as, to me it is just an incredibly beautiful theory that relates physics to geometry.

It is also fascinating that such a seemingly far-fetched theory can have such precise real-world predictions, as we’ll come to see.

General Relativity In a Nutshell: A Quick Overview

Here, I will present a quick overview of the whole theory of general relativity, which will lay down the most important ideas and help you see the big picture.

After that, we will look into each of these things in much more detail.

Now, you may have heard the phrase “gravity is the curvature of spacetime” at some point.

This phrasing can, however, often make general relativity seem like some sort of very abstract, almost science fiction -like topic.

General relativity, at its heart, is the generalization of the laws of physics to a universal (“covariant”) form, which is exactly what the complicated mathematics of general relativity do (this is explained more later).

The beautiful thing is that the mathematics also happen to have a geometric interpretation, which correspond to the notion of “spacetime being curved“.

Whether you wish to view general relativity in this geometric way is totally up to you.

The fact is, however, that the mathematics of general relativity give an accurate description of nature regardless.

In this article, we will, however, look at general relativity through a geometric lens, because it gives nice intuitive meaning to everything.

Down below I’ve collected the key concepts of general relativity, but for now, don’t get too hung up on the mathematical details, rather try to see the big picture.

• In relativity (both special and general), everything is described by something called spacetime. To put it simply, this just means that instead of thinking about time and the three spacial dimensions as separate things, we describe them by a four-dimensional spacetime. You can read my introduction to special relativity for more on this topic.

• To describe something in spacetime, you need to have a coordinate system (a time coordinate and spacial coordinates). Coordinate systems can be chosen freely, but universal laws of physics should not depend on any particular coordinate system (coordinate systems can still have a physical meaning, but a law of nature should be coordinate-independent because a law has to be the same for everyone). This is called the principle of general covariance.

• The laws of physics and gravity are described by tensors in general relativity. This is because tensors are mathematical objects that do NOT depend on any particular choice of coordinates, which means that they can be used to formulate physical laws in a universal way.

• The most important tensor in general relativity is the metric tensor. The metric tensor describes how distances are measured in a given spacetime and it is used to turn specific coordinates you use (the x’s in the equation below) into physical distances in spacetime.

• These distances (more accurately, the square of the distance) are, in their most general form, described by the following equation:

• An important principle underlying general relativity is the equivalence principle, according to which gravity and acceleration are completely equivalent, except for one thing; tidal forces. Tidal forces are something that can only appear due to gravity and they turn out to be the key thing in defining gravity in general relativity.

• Tidal forces in general relativity are described by a tensor called the Riemann tensor. The Riemann tensor also has a nice geometric meaning as describing the curvature of a space, which gives rise to the following interpretation; gravity corresponds to the curvature of spacetime.

• Objects under the influence of gravity move on geodesics through spacetime. A geodesic is essentially just a straight line, along which an observer does not experience any forces or acceleration. Geodesics can be calculated from the geodesic equation:

• This also describes how gravity is not actually a force. An object moves through spacetime in a straight line (geodesic), but if the spacetime is curved (gravity is present), this line will follow the curvature and may change direction (like if you were moving on the surface of a sphere, for example). This effect we call gravity. So, gravity is not really a force, but rather a manifestation of the structure of spacetime.

• Sources of gravity (spacetime curvature) are described by the Einstein field equations. These describe how things such as energy, momentum, pressure and stresses (i.e. the energy-momentum tensor) cause the curvature of spacetime:

Now, all of this may be a little overwhelming at first, but it will become more clear later when we discuss everything in much more detail and see just how all of these “pieces” fit together.

The Underlying Postulates of General Relativity

Every theory is always based on a few underlying principles or postulates, which pretty much lead to the specifics and the results of the theory. General relativity is no different.

Really there are two main postulates that general relativity is based on, which are the following:

• The principle of general covariance: this is mainly a mathematical assumption, yet it still does give some physical insights about how the laws of physics should behave. It also describes why the mathematical structure of general relativity happens to be what it is.
• The equivalence principle: essentially, this principle describes more of the physical assumptions that general relativity is based on and it can be thought of as a “physical restriction” to the principle of general covariance.

Combined, these two postulates provide the necessary underlying assumptions needed to build the framework of general relativity.

Down below, I’ll explain both of these in detail and as intuitively as possible.

To be fair though, there are a few more assumptions that need to be taken into account in order for general relativity to work the way it does (such as the assumption of a torsion-free connection).

These, however, are not necessarily considered as fundamental principles, but rather more technical assumptions, which we won’t get into too much here.

The Principle of General Covariance

The first important postulate or underlying principle of general relativity is called the principle of general covariance.

To be fair, this principle is actually not unique at all to general relativity, it is actually used in pretty much all of physics.

In short, the principle of general covariance states that laws of physics themselves should be independent of an observer’s motion, that is, they should have a covariant form. This means that a good law of physics should be built from mathematical quantities that are coordinate-independent, such as tensors.

Now, sometimes this principle is sort of taken for granted, because it sounds obvious. Of course, we want the laws of physics to be the same for everyone.

But, it does have a lot of important consequences and this principle is kind of at the base of all of physics, so it should be made clear here.

The word ‘covariant’ refers to a mathematical object whose components may change when you switch between different coordinates, but the object itself doesn’t.

Covariant quantities have a universal meaning, regardless of how we choose to represent them (we’ll talk about what this means when we get to tensors later).

In some sense, the principle of general covariance is only a statement about how the laws of physics should mathematically look like.

It does, however, also have physical consequences.

For example, it pretty much forces us to represent laws, such as the Einstein field equations, by using tensors.

This results in gravity also being affected by momentum and energy flow instead of just mass (energy density) like Newton’s law of gravitation states.

We’ll see exactly how this happens later on.

The Equivalence Principle

The second key postulate of general relativity is the principle of equivalence, which actually has several important consequences that are worth laying out here.

In short, the equivalence principle is one of the key postulates general relativity is based on and it states that the gravitational acceleration is the same for all observers in a gravitational field. This leads to gravity not actually being a force, but a property of spacetime itself (spacetime curvature).

Now, we already know from Newton’s law of gravity that all objects fall to the ground with the same acceleration (9.81 m/s² on the surface of the Earth).

This is due to the fact that the mass of the object itself cancels from Newton’s equation of gravity (or as sometimes stated, the inertial mass is equal to the gravitational mass).

This leads to a few important ideas that are at the heart of the equivalence principle:

• Since all objects experience the same acceleration due to gravity, irrelevant of the shape or mass of the object, it means that gravity acts very differently to other forces; for example, the electric force will depend on the charge of the object itself, but gravity is the same for each object.

• This can then be interpreted as gravity actually being an intrinsic property of spacetime itself. This logic is fundamentally where the notion of gravity being a geometric property of spacetime (“gravity=spacetime curvature”) comes from.

• Another statement of the equivalence principle is the fact that gravity cannot be distinguished from acceleration. A typical example of this is that imagine you’re in a closed box. It is impossible for you to determine whether gravity is pulling you down, keeping your feet stuck to the ground or whether the box is accelerating upwards. Thus, gravity and acceleration are equivalent (hence the name equivalence principle).

• There is, however, one exception to the above statement; tidal forces. Tidal forces come from the fact that objects may get stretched in a gravitational field because their different parts experience a different force. In simple accelerated motion, there is no such effect.

• Tidal forces turn out to be the key to defining gravity and spacetime curvature. They give a way to determine whether there is a real gravitational field (the spacetime is actually curved) or whether an object seems to be accelerating due to some kind of weird choice of coordinate systems (such as curved coordinates).

• Therefore, the meaning of spacetime curvature is really as simple as this; spacetime is curved if an object experiences real tidal forces and thus, there is a real, physical gravitational field present.

The above definition also has another very important consequence called metric compatibility. This is explained down below.

What Is Metric Compatibility and Why Is It Important? (click to see more)

The Mathematics of General Relativity Explained Intuitively

In this section, we’ll go over the important mathematical tools used in general relativity, such as the metric tensor and the Christoffel symbols.

We’ll also explore the physical and geometric interpretations of these.

It’s important, however, to realize that the focus of this article is not necessarily in the math, but rather what the math actually can tell us from a physics point of view.

For this reason, you do NOT have to pay much attention to mathematical details, such as some of the methods used to do tensor math, to understand this article.

What Are Tensors and Why Are They Used In General Relativity?

Essentially, tensors are the main mathematical objects used in general relativity simply because they have some very useful transformation properties.

The key idea is that a tensor is always the same in every coordinate system (in a technical sense, we say that a tensor transforms covariantly).

First of all, what is a tensor anyway? A tensor is simply a “collection of objects” (these objects are its tensor components) whose components transform in a nice way between coordinate changes.

Now, a mathematician may say that this is not an accurate enough definition, but for us, this works just fine.

Mathematically, a tensor is usually represented by some letter that has different indices (they can be upstairs or downstairs indices).

In general relativity, these indices are usually greek letters and they run from 0 to 3, which correspond to the different tensor components.

These components can be nicely represented as a “table” as follows (more accurately, this is called the “matrix representation of a tensor”):

T_{\mu\nu}=\begin{pmatrix}T_{00}&T_{01}&T_{02}&T_{03}\\T_{10}&T_{11}&T_{12}&T_{13}\\T_{20}&T_{21}&T_{22}&T_{23}\\T_{30}&T_{31}&T_{32}&T_{33}\end{pmatrix}

These T’s here are the components of this tensor T_µν. For example, T₀₁ is the component where µ=0 and ν=1.

For the context of general relativity and physics, here are some important facts about tensors:

• A tensor is always the same in all coordinate systems (i.e. tensors are coordinate-independent).
• The components of a tensor may be different in different coordinates (but the tensor as a whole isn’t, this property is called covariance). These components can, however, be mathematically compared in a fairly straightforward way.
• An equation involving ONLY tensor quantities is the same in all coordinate systems. Thus, the laws of physics should be these kinds of equations, in other words, tensor equations.
• Vectors and scalars are mathematically also considered to be tensors. Thus, they are also good for formulating the laws of general relativity.

Interestingly, the use of tensors and this property of covariance can even explain why specifically a moving charge creates a magnetic field. I discuss the physics behind this phenomenon in this article.

Now, the importance of tensors might not become clear before you actually start doing some calculations with them, so the bottom line for why tensors are used in general relativity goes as follows:

Tensors are commonly used in general relativity because they are coordinate-independent or covariant quantities. Since laws of physics have to be the same for everyone (they should have a coordinate-independent form), this means that also the laws of general relativity should be formulated from tensors.

This really comes down to the principle of general covariance, which was explained earlier.

Tensors allow us to formulate the mathematics of general relativity in a way that satisfies this principle.

Why Are Tensors Used in General Relativity, But Not in Newtonian Gravity?

Tensors, once you learn how to use them correctly, also greatly simplify calculations and equations.

Equations that would otherwise take up pages upon pages of very complicated formulas can be reduced to a simple and elegant form that can easily be worked with by using almost high school level math.

A very nice example of this is the Einstein field equations, which have a fairly nice and simple form when written using tensor notation. If you were to actually fully write out the field equations, they would look like a complete mess (you can see them fully written out on this page).

Metric Tensor: The “Fundamental Tensor” of General Relativity

Arguably the most important tensor in general relativity is the metric tensor, which I did mention earlier already. But what is the metric tensor actually and what is it used for?

In short, the metric tensor is used to define lengths and other geometric properties of spacetime in general relativity. The metric generalizes these properties to any curved space, so it can be thought of as the “measuring rod of spacetime”. Physically, the metric plays the role of the gravitational potential.

In some sense, the metric is only a mathematical tool since it cannot be measured directly.

This is just like the Newtonian gravitational potential cannot be measured, only the changes in it can; these changes, we call gravitational forces, however, it still has physical consequences.

Probably the most important use of the metric tensor in the context of general relativity is that it is used to calculate distances in a curved spacetime by defining a line element of the metric.

A line element can be simply thought of as a generalized Pythagorean theorem; it gives the square of a distance interval ds².

Line element of the metric gµν:

ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}

In the above formula (and throughout this article), we will be taking advantage of the Einstein summation convention.

This simply means that whenever there is an index repeated in both the upstairs and downstairs position, it is automatically summed over (from 0 to 3).

I’d recommend not getting hung up on these mathematical details.

I’ve written this article in such a way that you can understand the meaning of all the equations without knowing much about the math involved.

How Does The Above Equation Generalize The Pythagorean Theorem? (click to see more)

The metric tensor also has an interesting geometric interpretation.

Fundamentally, the metric is defined as the dot product between basis vectors:

g_{\mu\nu}=\vec e_{\mu}\cdot \vec e_{\nu}

This definition is not a very practical one in physics, but it does tell you about what the metric geometrically represents.

What Are Basis Vectors? A Quick Explanation (click to see more)

Let’s think about what the dot product really represents.

It essentially tells you how much two vectors are aligned or how much of the vectors are going in the same direction. I discuss this idea in great detail in the context of its physics in this article on the physical meaning of the dot product.

So, in other words, the metric tensor really gives a measurement of how much the basis vectors align with each other.

This is actually enough to define a notion of distance in any arbitrary coordinate system or spacetime, even a curved one.

Usually though, when we think of basis vectors, they are orthogonal to each other (they are at 90° angles to each other).

This definition really only works in “ordinary” space, which is not curved (Euclidian space) and more generally, in a curved space, the basis vectors may be pointing in totally arbitrary directions.

The metric tensor then simply gives a measure of how the basis vectors are aligned, which is enough to determine the geometry of the space itself.

The metric tensor (in 4-dimensional spacetime) can be represented as a 4×4 (symmetric) matrix as shown down below.

Typically in general relativity, the metric is written in terms of its line element, ds², by writing out the sum (see the formula from earlier). We’ll see how this works in practice later on.

g_{\mu\nu}=\begin{pmatrix}g_{00}&g_{01}&g_{02}&g_{03}\\g_{10}&g_{11}&g_{12}&g_{13}\\g_{20}&g_{21}&g_{22}&g_{23}\\g_{30}&g_{31}&g_{32}&g_{33}\end{pmatrix}

Components of the metric represented as a matrix (basically just a 4×4 “table”). Note that the metric is generally symmetric, meaning g₀₁=g₁₀, g₂₁=g₁₂ etc.

Practical Example: Metric Tensor and Line Element in Spherical Coordinates

The metric tensor is also used to define things like volume in curved spacetime, but this is really not too important right now.

Also, you may now be asking; sure, the metric tensor seems important in general relativity, but how do we actually find out what the metric is in a given spacetime?

The answer to this is by solving the Einstein field equations, which we will look at later on.

If you're interested to learn how the metric tensor is actually for physical applications, I'd highly recommend reading this article. In there, I discuss how the metric is used to analyze time dilation near a black hole, which actually has an extremely interesting geometric interpretation.

Christoffel Symbols & Their Geometric Meaning

The next important concept we’ll look at are the Christoffel symbols, which come up in the geodesic equation and in describing spacetime curvature (both which we will talk about in more detail later).

Christoffel symbols are mathematical objects that describe how basis vectors change in a coordinate system. In general relativity, Christoffel symbols describe changes in the metric throughout spacetime and therefore, they describe the acceleration experienced by an observer in a gravitational field.

There are essentially two ways to define the Christoffel symbols.

The first definition is a more geometric definition as the derivatives of basis vectors.

This helps to understand what exactly the Christoffel symbols tell us about the structure of a space itself (or more accurately, the structure of the coordinate system you happen to be using):

Down below, I’ve got a nice visual example of what these different Christoffel symbol components really represent.

The key point here is that the Christoffel symbols encode information about how the basis vectors change throughout spacetime.

This then is necessary to describe spacetime curvature (though to completely describe curvature, we’ll need a little bit more than just these Christoffel symbols).

What Do The Christoffel Symbol Components Geometrically Represent? (click to see more)

Now, the second way of defining Christoffel symbols can be done by using the metric tensor (this definition is much more useful in general relativity).

In other words, if we know the metric in a given spacetime, the Christoffel symbols can then be calculated by the formula given below.

Christoffel symbols in terms of the metric tensor:

\Gamma_{\mu\nu}^{\lambda}=\frac{1}{2}g^{\lambda\alpha}\left(\partial_{\mu}g_{\alpha\nu}+\partial_{\nu}g_{\alpha\mu}-\partial_{\alpha}g_{\mu\nu}\right)

Practical Example: How To Calculate Christoffel Symbols From The Metric (click to see more)

If you're interested, I actually have a whole article discussing various properties of the Christoffel symbols, which you'll find here. In there, I go into much more detail on the geometry of the Christoffel symbols as well as how to calculate them in practice through a little-known but extremely powerful method.

Now, some people like to think of the Christoffel symbols only as mathematical tools that are used in calculations without any real physical meaning.

This may seem so because they are not actually tensors, that is, they are not properly covariant like we would want the laws of physics to be.

This, however, is not necessarily true and the Christoffel symbols can actually be thought to have a physical meaning as well.

What Is The Physical Meaning of The Christoffel Symbols?

Christoffel symbols play a key role in the mathematics of general relativity, but do they have some kind of physical interpretation as well?

Physically, Christoffel symbols can be interpreted as describing fictitious forces arising from a non-inertial reference frame. In general relativity, Christoffel symbols represent gravitational forces as they describe how the gravitational potential (metric) varies throughout spacetime causing objects to accelerate.

To understand this a little better in the context of gravity, we need to first think of how a gravitational field is defined or how its effects can be measured in the first place.

A gravitational field is described by the acceleration (or force) an object would experience if it were to be placed in the field.

Now, Christoffel symbols, on the other hand, describe changes in basis vectors throughout a given coordinate system (as discussed earlier).

Physically, this actually corresponds to what are called fictitious forces (these are simply “forces” or acceleration effects that can only be observed in a specific frame of reference or coordinate system).

The interesting thing about this is that actually, gravity is a “fictitious force” in general relativity, which is caused by spacetime itself being curved in the presence of matter and energy.

Since the Christoffel symbols describe these “fictitious forces” (which are simply just the effect of a basis not being constant in some coordinate system), this means that Christoffel symbols play the role of describing how objects accelerate in a curved spacetime.

This indeed corresponds to the Christoffel symbols actually describing gravitational fields in general relativity. In fact, Christoffel symbols are gravitational fields in general relativity.

Now, why is this actually the case? In my opinion, the easiest way to see this is by comparing the definition of the Christoffel symbols in terms of the metric tensor to the definition of a gravitational field in Newtonian physics.

In Newtonian gravity, the gravitational field (i.e. gravitational acceleration at each point in the field) is represented by a vector field g. This g is defined as the negative gradient of the gravitational potential:

\vec{g}=-\vec{\nabla}\Phi=-\partial_x\Phi \vec e_x-\partial_y\Phi \vec e_y-\partial_z\Phi \vec e_z

Here, I’ve written out the gradient vector and the e’s denote the basis vectors in each direction (x,y,z).

Let’s now compare this to the definition of the Christoffel symbol in terms of the metric. The key idea here is to note that the metric itself represents the gravitational potential in general relativity.

The link between the Christoffel symbols and the Newtonian gravitational field is the weak-field limit, in which the Christoffel symbols actually reduce pretty much exactly to the Newtonian gravitational field.

This can be seen nicely by looking at geodesics in the weak-field limit and the corresponding Christoffel symbols, which you can read more about in this article.

Now, you may still be wondering; if the Christoffel symbols are not tensors, how can they represent something physical?

The answer is that while the Christoffel symbols are not covariant quantities (tensors), they can still have a physical meaning.

This idea is indeed quite puzzling, but essentially the answer is that in general relativity, gravity itself is a coordinate-dependent phenomenon (i.e. a “fictitious force”), but it still does have a physical meaning and so do the Christoffel symbols.

You’ll find more on this concept below.

How Can The Christoffel Symbols Represent Gravitational Fields Even Though They Aren't Tensors? (click to see more)

Now, it’s worth noting that you shouldn’t take this idea too literally all of the time. In many cases, especially in flat spacetimes, the Christoffel symbols won’t necessarily represent anything to do with gravity.

In these cases, the Christoffel symbols will generally represent some kind of fictitious forces induced by a non-inertial coordinate system.

For example, in a rotating coordinate system (but in flat spacetime), the Christoffel symbols actually have the physical interpretation of the centrifugal force and the Coriolis force, but nothing to do with gravity.

It is only in curved spacetime that the Christoffel symbols MAY happen to have a nice interpretation of a gravitational force.

The Covariant Derivative

Pretty much all the known laws of physics are described by differential equations. Arguably then, derivatives and differentiating vectors and tensors is very important.

But do derivatives work differently in a curved spacetime or in curved coordinates?

The answer is yes, which has to do with the basis vectors changing from place to place (if you have some arbitrary curved coordinates, the basis vectors will be different at different points).

A derivative describes the change in a vector (or tensor) with respect to some coordinate.

In a curved coordinate system, apart from the components of the vector changing because they are functions of the coordinates, the coordinates themselves may change.

To account for this, we use the so-called covariant derivative, which is the generalization of an ordinary derivative that works in ANY coordinate system (given that the coordinates are continuous).

Covariant derivative of a vector Aλ:

\nabla_{\mu}A^{\lambda}=\partial_{\mu}A^{\lambda}+\Gamma_{\mu\nu}^{\lambda}A^{\nu}

In general relativity, since we are dealing with curved spacetime and curved coordinates, the ordinary derivative is not very useful since the coordinates themselves are changing.

The covariant derivative fixes that since it takes into account these coordinate changes as well.

Therefore, we should use covariant derivatives if we’re dealing with curved spaces (although there are some exceptions where ordinary derivatives work fine in curved spaces too).

Where Does The Covariant Derivative Actually Come From? (click to see more)

In general relativity, the main use of the covariant derivative is that it’s a covariant version of the ordinary derivative (ordinary partial derivatives are NOT covariant in general).

This then means that it should replace ordinary derivatives when we’re dealing with curved spacetime.

In other words, to get properly covariant laws of physics, the process is really quite simple (most of the time); you just replace all ordinary derivatives with covariant derivatives.

A good example of using the covariant derivative to replace ordinary derivatives can be found down below.

Practical Example: Geodesic Equation From The Covariant Derivative (click to see more)

Curvature Tensors & How They Relate To Gravity in General Relativity

By now, it should be clear that gravity is described by spacetime curvature in general relativity.

The point of this section is to explain how spacetime curvature is actually described mathematically, which is by using different curvature tensors.

To give some insight into why the different curvature tensors have the form they have, it is important to understand that curvature fundamentally has to do with second derivatives of the metric.

The best way to intuitively think of this is by thinking of acceleration as the second derivative of position.

We already know that gravity is closely related with acceleration, so maybe gravity too has to do with second derivatives of something.

What makes this even more clear is that mathematically, the curvature of a function is also described by its second derivatives.

For example, the second derivative of a function f(x) being positive means that the function will “curve” upwards.

Now combine these with the notion of spacetime itself being described by a metric tensor.

This very strongly suggests that, indeed, spacetime curvature (and thus, gravity) should be described by second derivatives of the metric, which is what the different curvature tensors actually do include.

Down below, we will also take a look at what exactly the different types of curvature tensors geometrically (and also physically) represent, which should give some very interesting insights into these tensors.

We’ll also see how all of these will piece together to produce a beautiful theory of gravity.

The most important tensors to describe spacetime curvature are the following ones:

• The Riemann tensor: this tensor gives a complete description of any curved space and is really all that is needed to describe spacetime curvature completely. The Riemann tensor is built out of derivatives and products of Christoffel symbols (or written in terms of the metric, basically second derivatives and “squares” of the metric):

• The Ricci tensor: this tensor is built from the Riemann tensor, but it contains less information about the curvature. This tensor is, however, needed for general relativity and the Einstein field equations. Physically, the Ricci tensor describes how the spacetime volume of an object changes due to tidal forces (spacetime curvature). The Ricci tensor is mathematically defined as the contracted Riemann tensor:

• The Ricci scalar: this tensor is actually a scalar (as the name may suggest), which is just a single number. Physically, the Ricci scalar describes the total change in the spacetime volume of an object (but does not give any information about the change in each direction specifically unlike the Ricci tensor). The Ricci scalar is the contraction of the Ricci tensor:

• The Einstein Tensor: this tensor shows up in the Einstein field equations and it is useful for simplifying the equations a little bit. The Einstein tensor does not have any particularly important physical meaning and in some sense, it is just a convenient simplification. The Einstein tensor is defined in terms of the Ricci tensor and the Ricci scalar:

G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R

It’s worth noting that there are plenty more of these curvature tensors, such as the Weyl tensor and the Kretschmann scalar (which I’ll briefly explain later).

The tensors given above, however, are the most important ones for understanding the basics of general relativity.

The Riemann & Ricci Tensors: What Do They Represent Geometrically and Physically?

The first and probably the most important tensor to describe the curvature of spacetime is the Riemann curvature tensor.

The Riemann tensor is enough to fully determine how any space is curved, but what does it actually represent geometrically?

Geometrically, the Riemann tensor completely describes the curvature of any space and it represents how the components of a vector change due to curvature as it gets parallel transported around a loop. Physically, the Riemann tensor represents the effects of tidal forces due to gravity in general relativity.

Visually, there is quite a nice way to see the geometric meaning of the Riemann tensor.

Imagine we have a vector that we move around some small loop in a curved spacetime (its length has to stay fixed, this is a fundamental physical requirement!).

Below I have a picture of this in two dimensions (since I can’t actually draw a four-dimensional spacetime).

The Riemann curvature tensor then describes how the components of this vector change due to curvature (this is, in fact, enough to fully describe the curvature of any space).

Down below you’ll find a more mathematical way to see where the Riemann tensor actually comes from, which should be enlightening.

Essentially, it is done by using covariant derivatives and a geometric method called parallel transport.

Also, physically the Riemann tensor corresponds to the tidal tensor in Newtonian gravity.

It is, in fact, the general relativistic equivalent of this tensor (this is explained in more detail later).

This means that the physical content of the Riemann tensor is that it gives a complete description of tidal forces in general relativity.

This relates right back to the beginning of this article, where we talked about tidal forces being the key to describing spacetime curvature.

Derivation of the Riemann Tensor From Parallel Transport (click to see more)

The Ricci tensor and the Ricci scalar, on the other hand, have to do with changes in spacetime volume and they both have a very nice geometric interpretation.

The geometric interpretation of the Ricci tensor is that it describes how much a volume between a group of geodesics changes in each direction due to curvature as one moves along these geodesics. The Ricci scalar describes the total volume change but doesn’t give information about any particular direction.

In a very rudimentary sense, you can think of it like this; imagine you have a group of geodesics (essentially lines in spacetime).

These geodesics will then enclose some kind of volume in spacetime.

Geometrically, the Ricci tensor then describes how much this spacetime volume changes as you move along these geodesics.

The different components of the Ricci tensor correspond to the volume changes in different spacetime directions.

If you're interested, I actually have a whole article on the Ricci tensor where I discuss its geometric and physical meaning as well as some of its properties. I also include a bunch of examples of different Ricci tensors in different spacetimes.

Now, while I can’t really draw a proper four-dimensional spacetime, it’s still possible to visualize this in two dimensions, for example.

Imagine we have a two-dimensional spacetime with two geodesics that enclose a volume between them (practically it’s an area since we’re in two dimensions, but imagine it as a volume):

Now, the Ricci tensor is useful for describing curvature mathematically in general relativity, but does it also have a specific physical meaning?

In short, the physical meaning of the Ricci tensor is that it describes how the spacetime volume of an object changes along a geodesic. Physically, the components of the Ricci tensor can be thought of as describing how much the spacetime volume of an object changes in each direction due to tidal forces.

Let’s think of this in more detail. In particular, let’s think of a physical object with some well-defined volume (a ball for example).

In spacetime, all the different parts of the object will follow their own geodesics through spacetime (essentially, you can think of every atom of an object following its own spacetime geodesic).

Therefore, if curvature is present, these geodesics may begin deviating from one another and the spacetime volume between them will change.

Physically, this means that the object will get stretched and squeezed in different directions.

This squeezing and stretching, on the other hand, corresponds to the effect of tidal forces, which are simply the result of geodesic deviation (I’ll explain this in more detail later on).

So, the physical description of the Ricci tensor is literally how much the volume of an object changes due to gravitational tidal forces.

Also, note that the object gets deformed in spacetime, not just in the usual three-dimensional space, so the object will also get stretched and squeezed in time, which is called time dilation.

Similarly to the Ricci tensor, the Ricci scalar also has a clear physical meaning.

The physical meaning of the Ricci scalar is that it describes how much the spacetime volume of an object moving along a geodesic changes due to tidal forces. Unlike the Ricci tensor, the Ricci scalar only describes the total change in volume, not the change in any particular direction in spacetime.

The Energy-Momentum Tensor

As you may know by now, gravity is described by spacetime curvature. But according to Newtonian physics, gravity is caused by mass, such as the mass of a planet.

So far, we have not talked about what actually causes gravity or the curvature of spacetime in general relativity.

The answer is a bit more complicated in general relativity than just mass.

According to special relativity, mass is just another form of energy (from E=mc², which you can read more about in this article like where this famous equation really comes from).

This then suggests that gravity should really be caused by all types of energy and not just this one special form of energy called mass.

Not only that, but it turns out that gravity is also caused by energy fluxes as well as momentum fluxes.

These are all describes by one mathematical object called the energy-momentum tensor T_µν.

The energy-momentum tensor tells us how four-momentum flows through spacetime.

Four-momentum in special relativity is a vector that combines both energy and momentum, which you can read about in this special relativity article if you’re not familiar with the concept.

The different components of the energy-momentum tensor therefore describe how energy and momentum “flow through time” (= energy and momentum density).

The energy-momentum tensor also describe how they flow through space (= energy flow, which is simply just momentum and momentum flow, which is closely related to forces like pressure and shear stress).

The components of the energy-momentum tensor can be nicely represented as a matrix:

Here, I’ve written the components by using the coordinates (t,x,y,z) instead of just (0,1,2,3) to make the meaning of these components easier to understand intuitively:

Each of these components above can contribute to the curvature of spacetime (through the Einstein field equations) and thus, can cause gravity.

A noteworthy fact is that usually, the T₀₀ -component (energy density, i.e. mass) is MUCH larger than any of the other components.

This may indeed be where Newton’s observations of mass being the cause of gravity, come from.

It simply wasn’t possible to deduce these other gravitational causes by just observing our solar system, so Newton did not include these in this gravitational law.

In most cases, these shear stress components will be zero (since it’s common to study objects that could be considered as “perfect fluids“, which don’t have any internal stresses).

The pressure components may sometimes play a role if we’re studying things like stellar interiors (the structure of stars, for example).

Practical Example: Energy-Momentum Tensor of a Perfect Fluid (click to see more)

Geodesics & How Objects Move in Curved Spacetime

We are now ready to actually get to the interesting physics concepts of general relativity.

The key idea is the notion of geodesics, which are in their most simple sense, just trajectories through spacetime.

Fundamentally, a geodesic is just a straight line. If you happen to be in a curved spacetime, however, straight lines will naturally follow the geometry of that spacetime, so the trajectory is not really straight even though it may appear so for an observer moving along that geodesic.

The best way to intuitively understand this is by imagining you’re on a sphere (surface of the Earth, for example).

Now begin walking straight ahead without turning at any point.

Eventually, you’ll still end up where you started (you’ve looped around the whole sphere) even though you walked in a “straight line” all along.

Thus, the effect of curvature will determine how a straight line trajectory actually looks like (in this case, curvature of the sphere).

To you, it seemed as though you were heading on a straight line all along, but clearly if you went around the sphere, the trajectory was not really straight.

Now, how does all this relate to physics? The best way to understand this is to think about what a “straight trajectory” actually means physically.

According to Newton’s laws, an object will always move in a straight line if it experiences no acceleration.

But, because the “straight line” actually did not remain straight, does this mean that there is really an acceleration?

To understand this, we will look at an object moving in curved spacetime that does not experience any acceleration.

Mathematically, no acceleration means that the velocity is a constant and its time derivative has to be zero.

In relativity, this corresponds to the derivative of four-velocity (velocity in spacetime) with respect to proper time (the “universal” definition for time, which is invariant) being zero:

\frac{d\vec u}{d\tau}=0

Four-velocity is typically denoted by the letter u.

Now, any vector can be expressed as the sum of its components with the basis vectors (you may be familiar with this in typical Cartesian coordinates already).

So, we can say that the four-velocity is simply u^µe_µ (summing over µ from 0 to 3).

Here comes the important part; if spacetime is curved (which we’re interested in right now), the basis vectors may not be constant.

Therefore, we have to use the product rule:

\frac{d\left(u^{\mu}\vec{e}_{\mu}\right)}{d\tau}=0\ \ \ \Rightarrow\ \ \ \vec{e}_{\mu}\frac{du^{\mu}}{d\tau}+u^{\mu}\frac{d\vec{e}_{\mu}}{d\tau}=0

Here we have the derivative of a basis vector, which suggests that maybe this has something to do with Christoffel symbols.

Indeed, with a little bit of math, we can express this by using Christoffel symbols and we then end up with the geodesic equation.

The geodesic equation:

\frac{du^{\mu}}{d\tau}=-\Gamma_{\lambda\nu}^{\mu}u^{\lambda}u^{\nu}

Now, what is the actual physical meaning of this? The left-hand side looks very much like an acceleration term, but what does the stuff on the right represent?

This stuff comes from the fact that our spacetime was assumed to be curved (the basis vectors may not be constant).

Because of this, an additional “force term” appears on the right-hand side, which causes an acceleration.

This seems weird, because we started by looking at an object that did not experience any acceleration.

We therefore see that if an object is moving in curved spacetime, it will undergo an acceleration caused by the curvature itself, even if no forces are present.

An important property is that the object itself does not experience any acceleration, but any outside observer would see the object accelerate.

This is a key property of gravity (the fact that you would not feel any acceleration if you’re just “free-falling” in a gravitational field, i.e. following a geodesic, but anyone looking at you would clearly see you accelerate).

This now has a clear explanation in terms of general relativity and spacetime curvature.

This is fundamentally why gravity is NOT a force, but rather a property of spacetime itself, namely the curvature of spacetime.

This also gives a clear definition for what geodesics are:

A geodesic is a trajectory, along which an object does not experience any acceleration. All objects will naturally move along geodesics through spacetime and geodesics in curved spacetime are equivalent to motion under the influence of gravity.

There is actually another interpretation for what a geodesic is and it is that a geodesic is the shortest distance between two points in spacetime.

In flat spacetime, this is naturally just a straight line (corresponding to no gravity), but in curved spacetime, the shortest path is some curve which corresponds to motion in a gravitational field.

Geodesic As The Shortest Distance Between Two Spacetime Points (click to see more)

Now that we’ve established what geodesics are, let’s look at the geodesic equation in more detail:

In a more practical sense, the geodesic equation is essentially the law of motion in general relativity (just like Newton’s second law, F=ma, is in Newtonian physics).

This can be quite easily seen if we express the force in Newton’s law as the gradient of the gravitational potential and the ma -term as mdv/dt (we can also multiply the geodesic equation by mass m on both sides):

Also, the way you practically use the geodesic equation is absolutely equivalent to Newton’s law:

• The motion of any object under the influence of gravity is governed by the geodesic equation in general relativity, just like it is governed by Newton’s second law in Newtonian physics.
• In general relativity, you specify the metric tensor (gravitational potential) and the geodesic equation tells you the trajectory of an object (for example, the orbit of a planet). In Newtonian physics, you specify the gravitational potential and Newton’s second law tells you the trajectory.

You’ll find an example of how the geodesic equation is practically used down below.

Practical Example: Geodesics on a Sphere (click to see more)

Geodesics describe pretty much everything that moves under the influence of gravity, including massless photons or light. In case you're wondering how light can possibly be affected by gravity and how the geodesic equation leads to deflection of light near a star, I recommend reading this article.

Tidal Forces in General Relativity

The effects of tidal forces can be seen as the rise of sea levels caused by the Moon's gravity.

However, tidal forces are always present when an object is in a gravitational field and these may cause the object to get deformed.

Newtonian gravity explains this by the fact that different parts of an object experience a different gravitational force as described by Newton's law of gravity, F=GMm/r².

General relativity, on the other hand, can explain the phenomenon of tides as well.

Tides are explained in general relativity by the fact that different parts of an object each follow their own geodesics through spacetime. These geodesics may begin deviating from one another due to curvature (gravity), causing the object to deform, which can then be physically observed as tidal forces.

In my opinion, this explanation is extremely elegant, but it can be a bit difficult to conceptualize.

The key idea is that everything in spacetime moves along geodesics.

If the spacetime is curved (gravity is present), these geodesics correspond to trajectories of objects under the influence of gravity.

Moreover, if everything in spacetime follows its own geodesic, it means that different parts of an object will each have their own geodesic.

Let's look at an example to visualize this; say we have a person named James, for example, who is in a gravitational field (in a curved spacetime).

In this gravitational field, both James' feet and upper body will each follow their own geodesic, which are determined by how the spacetime is curved..

Technically, each atom in James' body will have its own geodesic, but we'll look at this in a more macroscopic context.

Mathematically, the effects of these tidal forces in general relativity are described by the equation of geodesic deviation.

This, as the name may suggest, describes how two geodesics deviate from one another due to spacetime curvature.

The equation of geodesic deviation:

\frac{D^2\epsilon^{\mu}}{D\tau^2}=R_{\rho\lambda\nu}^{\mu}u^{\rho}u^{\lambda}\epsilon^{\nu}

Now, what do all these things mean? Let's dissect this equation and see what it really says:

Visually, this separation vector describes how two geodesics are separated (each component of this vector tells you about each spacetime direction).

This length and direction of this vector may change as you move along these geodesics:

Interestingly, tidal forces are also described by a tensor in Newtonian gravity, but it does not give a relativistically valid description.

In fact, the Riemann tensor turns out to be the relativistic extension of the Newtonian tidal tensor.

Tidal Forces in Newtonian Gravity vs in General Relativity (click to see more)

Another important mathematical object related to tidal forces is the Weyl tensor.

The math regarding the Weyl tensor can get quite complicated, but essentially the Weyl tensor describes how the shape of an object gets deformed due to tidal forces.

In contrast, the Ricci tensor only describes the change in an object's volume (while not giving any information about changes in shape).

The Weyl tensor, on the other hand, describes the changes in shape, but doesn't give any information about how the volume changes.

In fact, combined together, the Ricci tensor and the Weyl tensor give a complete description of tidal effects.

Together they are equivalent to the Riemann tensor, in that they describe spacetime curvature and tides completely.

The usefulness of the Weyl tensor really comes in when studying gravitational radiation.

These can cause squeezing and stretching effects on gravitational wave detectors, for example (i.e. deform the shape of the detector).

The Einstein Field Equations: What Actually Causes Spacetime Curvature?

We've pretty much covered the most important concepts of general relativity so far.

There is just one more question left; by now, you know how curvature is described and how it causes the effects of gravity, but what causes spacetime curvature in the first place?

The answer to this is simple; the energy-momentum tensor!

In other words, the energy-momentum tensor acts as a source of gravitational fields, which means that spacetime curvature can be caused by, not only mass, but all types of energy and energy fluxes, momentum, pressure and different kinds of stresses.

How exactly the energy-momentum tensor acts as a source is described by the somewhat famous Einstein field equations.

The Einstein field equations:

G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

Note; if you want to see where exactly the Einstein field equations come from, you can check out this article, in which we derive the field equations step-by-step. You can also get the article as a PDF here.

At first sight, this equation may seem fairly simple. Don't be mistaken though, it is actually extremely difficult to solve in reality.

Moreover, this really represents multiple equations, since there is one for each combination of µ and ν (there is actually 10 independent equations in total contained in these field equations).

To get an idea of how complicated these field equations actually are, you can see them fully written out in terms of the metric (and its first and second derivatives) on this page.

Now, the left-hand side of the field equations contain the "curvature part" which is the Einstein tensor.

Remember from earlier that the Einstein tensor is a combination of the Ricci tensor and Ricci scalar, which contain a lot of Christoffel symbol terms.

The right-hand side is the energy-momentum tensor (which may or may not have a complicated form, it depends on the problem).

It is also combined with some constants (G is the gravitational constant and c is the speed of light).

A Comparison of the Einstein Field Equations To Newton's Law of Gravity (click to see more) What Do These Constants In Einstein's Field Equations Mean? (click to see more)

What Are The Einstein Field Equations Used For?

At the most basic level, the Einstein field equations are just that, equations. But equations for what? Equations for the metric, of course!

The Einstein field equations describe how energy and momentum cause curvature and curvature itself is described by the different curvature tensors, which are really built from the metric.

In other words, the Einstein field equations determine what the metric looks like for a given spacetime, depending on the energy content of that spacetime.

To get the metric tensor, you have to then solve the field equations.

Now, the problem with this is that the field equations are a set of extremely complicated, non-linear second order differential equations, which in most cases, don't even have an exact solution.

You therefore have to either use a computer and solve them numerically or make some simplifications.

The most common simplifications and the solutions obtained from these are:

• Vacuum solutions: these describe a spacetime which contains no matter (T_µν=0), but the spacetime can still be curved. Examples of such solutions are the Schwarzschild metric (describing spacetime outside of a static, spherically symmetric object, like a planet) and the Kerr metric (which describes spacetime around a rotating object, such as a rotating black hole).

• Weak-field solutions: these describe a spacetime where gravity is "weak" (the metric only deviates slightly from the flat spacetime Minkowski metric, so it's of the form g_µν=η_µν+h_µν where h_µν describes small perturbations in the metric). This type of formalism is also called linearized gravity and it is used to study gravitational waves as well as relate general relativity to Newtonian gravity.

• Symmetric solutions: these describe spacetimes that have a large degree of symmetry, making the field equations much much simpler. Examples of this include spherical symmetry (Schwarzschild metric) and homogeneity or isotropy (such as the Friedmann-Robertson-Walker metric, which describes things like the expansion of the universe).

So, in a nutshell, the process for solving a problem in general relativity usually goes more or less like this:

1. Specify a set of symmetries or simplifications, such as the ones given above. This should reduce the complexity of the Einstein field equations.
2. Specify the energy and momentum distribution in your spacetime (i.e. the energy-momentum tensor) if not done already in the previous step.
3. Calculate the Einstein tensor by using the Einstein field equations and the simplifications you specified. Note that you'll have to calculate the Riemann tensor first and from that, the Ricci tensor and scalar, which form the Einstein tensor.
4. From the Einstein tensor, you can solve for the metric (either numerically or analytically, if possible).
5. Once you have the metric, you have a complete description of your given spacetime. You can calculate pretty much anything you wish to (such as the four-velocities, four-momenta or geodesics of objects in that spacetime).

If you're interested, I have an article discussing time dilation near a black hole, which makes use of the Schwarzschild metric that can be used to describe a black hole. I also discuss some other metrics, such as the Kerr metric, which is also a solution to the Einstein field equations.

For another application, I suggest reading this article on orbits around black holes, which covers how to use a so-called effective potential to intuitively visualize these black hole orbits. For more mathematical details on this concept, I recommend this article on orbits of light.

Also, this article on the Friedmann equations looks at how the Einstein field equations are used to derive arguably the two most important equations in cosmology and how these lead to predictions like the expansion of the universe.

The Principle of Least Action in General Relativity

So far, I haven't really explained where exactly the Einstein field equation comes from and why it has the specific form it has.

This is what we'll do now and the best way to do this is by using the principle of least action, which is one of the most fundamental concepts in all of physics.

If you're not familiar with the principle of least action (technically, it's the principle of stationary action), I recommend reading my article on Lagrangian mechanics (Lagrangian mechanics is a formulation of mechanics that uses this principle).

Essentially, the basic idea of how this principle is used in physics is this: pretty much all theories of physics each have their own action integral.

The action is usually found simply by trying out what works and what doesn't (typically, there are really only a few options since there are some fairly strict requirements as to how the action should look like).

Then, the action has to be made stationary (or minimized, as the name, principle of least action, suggests).

This is done mathematically by varying it and setting this variation equal to zero.

Now, what's the point of doing all this?

Well, once you calculate the variation and set it equal to zero, you then have the fundamental equations for that given theory.

In the case of general relativity, this leads to the Einstein field equations (which we otherwise couldn't derive from anything else, they would simply have to be assumed).

The Einstein-Hilbert action:

A=\frac{c^4}{16\pi G}\int_{ }^{ }R\sqrt{-g}d^4x

The action that leads to the field equations is called the Einstein-Hilbert action.

This action is actually the simplest possible action we could construct that has all the properties we wish.

These properties include:

• The action has to be an integral over four-dimensional spacetime, d⁴x, since that's what we're interested in in general relativity.
• The action should somehow involve curvature, since we've determined that gravity has to do with curvature of spacetime (which is why the Ricci scalar R is there; it's the simplest possible curvature quantity we could have).
• The action has to be an invariant quantity or simply a scalar/scalar function (meaning it has to be a scalar, which is why there is this weird square root term; it turns out that, in curved spacetime, integrals over a spacetime volume, d⁴x, have to be multiplied by such a term involving the determinant of the metric, g, otherwise they would not be invariant).

Mathematically, varying this and setting it to zero (i.e. applying the principle of stationary action) means the following (the variation we denote by a δ-symbol, but it's really closely related to derivatives):

\delta A=\frac{c^4}{16\pi G}\delta\int_{ }^{ }R\sqrt{-g}d^4x=0

What you'll get as the end result, are exactly the Einstein field equations:

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=0

These are the vacuum equations where T_µν=0; it's also possible to derive the full field equations by using this principle.

The mathematical steps to do this are quite complicated but if you're interested in seeing exactly how the field equations are derived from this action principle, this is indeed one of the two ways I derive them in my article on the full derivation of Einstein's field equations.