Recommended : Manifolds and Curvature
The purpose of this article is to introduce the basic concepts of what GR is about, without getting lost in mathematics, but also without succumbing to the “easy way out”, as given by seemingly simple yet highly misleading visualisations such as the ever so popular “rubber sheet analogy”. It is to be understood as a plain text, “first primer” type of introduction. This article is aimed at the interested lay audience with little or no mathematical background, so I will sacrifice mathematical rigour for clarity and understanding; the mathematical side of GR will be presented in a future article. While the finer details of GR are undeniably hard, its basic principles are simple, and can be understood by anyone.
Let us first of all recall a few important points from our discussion of Special Relativity – if you haven’t read the articles linked to above, I would urge you now to do so :
- An event is a point in space at an instant in time
- Space-time is the collection of all events
- The geometry of space-time is the relationship between events, i.e. of the relationship between measurements of time and space taken at different places and times
- In the absence of gravity, space-time is flat, and is described by the laws of Special Relativity
- Time and space have no universal meaning, they are observer-dependent concepts
Special Relativity is a model that describes situations where gravity is either absent, or can be neglected; the aim of General Relativity is to relax this restriction, and incorporate gravity as well into our space-time framework.
Since space-time is the set of all events, it is fundamentally a static concept – it provides a description of all events regardless of where and when they are. Of course, this isn’t how we as human beings actually experience the world – for us, the universe isn’t static, it is a dynamic entity full of motion, and systems evolving in time. It still has four dimensions of course, but we are only ever able to perceive a succession of instants in time; we cannot freely move in time in the same manner as we can move in space. This is a peculiar property of the real world, for which the theory of relativity does not offer an explanation. In fact, how the arrow of time arises, why it has a preferred direction ( the future ), and why we see only a succession of moments, is still an open question in physics in general, not just in classical relativity.
To describe the dynamics of moving particles, in relativity we use the concept of world lines. These are just all the spatial positions of a test particle / object taken at each corresponding instant in time. As such, a world line is again a static construct; for example, the moon with respect to the earth would be a periodic elliptic trajectory in space, but a static helix in space-time  :
When analysing world lines, we find that not all of them are created equal. Of course, all bodies and test particles trace out some form of world line, but there is one very special type – that traced out by bodies not subject to any external force.
Have you ever jumped off a high enough board and into a swimming pool ? Did you notice that “butterfly” feeling you got in your stomach after you left the board, but before you hit the water ? That feeling was the absence of any force acting on you; when you were in free fall, you were weightless. Your body acted as a kind of accelerometer, and felt the absence of the downward acceleration we usually experience while standing on the Earth’s surface; in free fall, an accelerometer reads exactly zero. This means that in free fall, there is no force acting at all on a test particle ( remember F=ma, with a=0 ).
Free Fall. Free fall is the simplest and most natural state of motion for all objects; an accelerometer co-moving with a freely falling test particle reads exactly zero at all times, so there are no external forces acting on the test particle whatsoever. A free fall frame is locally an inertial frame, because there is no proper acceleration.
Now think back to what we have already learned in our articles on Special Relativity – the geometry of space-time is such that an inertial observer on whom no forces act ( and hence who experiences no proper acceleration ) traces out the longest proper time between any two given events. This also holds true in free fall in general, and it gives rise to a fundamental principle which governs how test particles move :
The Principle of Extremal Ageing. A test particle in free fall will trace out a world line that maximises its proper time. These very special world lines are called geodesics of space-time.
Remember here the meaning of “proper time” – it is just precisely what a clock that travels along that world line physically reads. As such, it is also equal to the geometric length of the world line segment in space-time.
If there is no gravity at all, as in the flat space-time of Special Relativity, “free fall” just means that our test particle is purely inertial – it experiences no acceleration, and no forces act on it at all. The geodesics of flat Minkowski space-time are hence perfectly straight lines.
So, in the theory of relativity, the driving influence of all dynamics is the fact that test particles always age into the future. They cannot do anything else – you can stand still in space, but you cannot stand still in time. Everything always keeps ageing into the future. Now suppose that you have some massive body, such as the planet Earth, and you place an accelerometer ( initially at rest ) somwhere close to it in free space, so that no external forces act on it; the accelerometer will function as our test particle. What happens when the accelerometer ages into the future ? Intuitively, we know the answer – it will slowly start falling towards the Earth, under the influence of gravity, but it will keeping reading zero at all times, since it is in free fall. Our test particle falls towards a central body under the influence of gravity as it ages into the future, yet it detects no force acting on it at all ! Gravity, therefore, cannot be a force in the old Newtonian sense.
Let us examine the motion of test particles near a massive body more closely, and see if we can deduce a pattern here. We shall assume for simplicity that our massive body does not rotate, and has no net electric charge. We already know that the world lines of such test particles obey the principle of extremal ageing, so they are geodesics of space-time; however, they are obviously not straight lines, since these particles initially move slowly, and then start going faster and faster as they approach the central body in free fall. Plotted onto a space-time diagram, you would get some kind of curve. Consider now the case of three test particles, stacked vertically above a central body, like so  :
Again, we can take our test particles to be accelerometers, and again we would find that they all read exactly zero at all times, indicating that no force acts on any of them. As these test particles age into the future, this is what happens  :
As expected, as time passes, they start approaching the central body under the influence of gravity, and they do so in such a way that the distance between the test particles increases, while their distance from the central body decreases. The world lines of our particles start to curve and diverge – this “fanning out” can be understood as a relative acceleration between test particles. The word relative is important here, because none of the accelerometers records anything – the distance between the particles increases, even though no force is detected to be acting on them. Relative acceleration ( as opposed to proper acceleration, which would be detectable on our accelerometers ) between test particles in free fall is called geodesic deviation, meaning that the geodesics these particles trace out over time deviate from one another.
So far so good. Now let’s see what happens if we stack the particles not vertically, but horizontally above the central body, like so  :
The initial setup is similar, with the only difference being the arrangement of the test particles. As they age into the future, this is what happens  :
Again, we find that geodesic deviation happens – the particles experience relative acceleration towards the central body, but they also experience relative acceleration towards one another. Unlike in the first case ( vertical stacking ), the particles are now getting closer to each other as time goes by, they converge  :
Physically speaking, this is just what we mean by tidal gravity – test particles always fall towards the centre of a massive body, and if these particles were initially horizontally separated, then that means they will get closer as they fall. Again, no detectable forces act on them.
This nicely covers motion outside the central body, i.e. outside the Earth, in the vacuum of free space. But what about the interior of Earth ? Suppose we drill a shaft all the way through the planet – how do test particles move in the interior ? Let’s set up an experiment like this  :
We assume that the test particles don’t interfere with each other as they move. What we find is that the particles, initially at rest, take about 21min to fall into the centre, and then another 21min to come at rest again at the opposite side of Earth. This is true wherever we start the experiment – if we start from rest half way down the shaft, the particles will still take 42min to do their thing. But what about the world lines of these particles ? Well, let’s plot them onto a diagram  :
And again we find the same phenomenon – the world lines of the particles are geodesics, and under the influence of gravity they experience relative acceleration, i.e. geodesic deviation. And again, accelerometers placed with the particles read zero at all times, so there are no forces acting. Because under none of the above circumstances we were able to detect any forces whatsoever, we are left with only one conclusion :
Gravity. Gravity is not a Newtonian force, but geodesic deviation ( = relative acceleration ) between the world lines of test particles.
To make the idea of geodesic deviation intuitively clear, consider what happens on the surface of our Earth  :
As A and B are moving north, the geodesics they are on ( which are great circle segments on a sphere ) will converge more and more, until they eventually intersect at the North Pole. That means that A and B get closer together as they move north, not because there is any kind of force acting between them, but purely because of the geometry of the surface they are on.
Let us summarize our findings here :
- Test particles in free fall ( i.e. without external forces acting on them ) trace out world lines that maximize proper time. These are called geodesics in space-time
- In the absence of any sources of gravity, geodesics are straight lines in space-time
- If there are sources of gravity present ( such as a planet ), geodesics cease to be straight everywhere – they diverge and converge, and describe “curves”, as in the above drawings
- Gravity is geodesic deviation of world lines
Point (1) can be empirically tested by comparing a clock in free fall to a clock that experiences proper acceleration, for example one onboard a rocket ship. A similar experiment readily confirms (2). Point (3) is likewise confirmed by experiment and observation of freely falling bodies and their trajectories over time. Definition (4) then follows as a consequence of (1)-(3). Taken together, these are all not just theoretical conjecture, but a description of empirical findings. Well, I must of course qualify this a bit – we haven’t drilled any shafts through Earth, so the behaviour of test particles in the interior of the Earth is an extrapolation. Nonetheless, this could in principle be tested in a short segment of shaft, perhaps just a few kilometers deep, so we’ll put it down as an empirical finding as well. All of these taken together point to a simple, yet very profound conclusion :
Space-Time and Gravity. Because geodesics are not straight lines in the vicinity of sources of gravity, as evidenced by simple observation, the geometry of space-time cannot adequately be described as the flat Minkowski space-time of Special Relativity. Instead, we find that the relationship of events near a gravitating object is the same as the relationship between points on a manifold with intrinsic curvature. The Theory of General Relativity provides a description of gravity based on this model of curved manifolds. Gravity therefore becomes a geometric property of space-time.
You will find that almost all pop-sci sources, as well as most textbooks, just state that “space-time is curved in the presence of sources of gravity”. However, I prefer to state things like I did in the above – that the relationship between events ( remember that world lines / geodesics are just a sequence of events ) near a gravitating body is the same as the relationship between points on a manifold with curvature. Curved space-time ( = curved manifolds ) is a description, a model, of how test particles behave in the vicinity of gravitational sources. General Relativity is a map, and the universe is the territory – there is a definite and very explicit relationship between the map and the territory it describes, yet the two are not actually the same. This is true not just for GR, but for all models of physics. Space-time curvature is very real in the sense that it accurately models the motion of test particles, but it is not real in the sense of space-time being some kind of mythical “substance” that has mechanical properties, and can be distorted and curved by mechanical action and forces somehow acting from “the outside”. That is most emphatically not what General Relativity is about – we aren’t dealing with an updated version of the old luminiferous ether here, but instead we are examining how measurements taken at different places and times are related to one another. It’s about geometry, not mechanics.
Space-Time Curvature. Space-time curvature is to the universe as the map is to the territory. There is a well defined relationship, but not an identity. Space-time c urvature concerns itself with how measurements of space and time taken at different places and times are related to one another; it has nothing whatsoever to do with mechanical distortions of some ethereal “fabric”. Space-time is not a substance with mechanical properties ! Nonetheless, space-time curvature is a real phenomenon, and not just an artefact of mechanical changes in our measurement instruments, or pure mathematical conjecture.
General Relativity has nothing to say as to what space-time “actually is”; it provides only a description of the macroscopic, classical behaviour of test particles, and how they are related to sources of gravity, and as such it is an extremely successful model of physics. However, GR does not tell us anything as to the miscroscopic structure or “true nature” of space-time, or how it arises in the earliest moments of the universe, just like a topographic map may depict the structure of a territory without being able to explain the geological processes that lead to the formation of same, or the nature of the rocks that make up the mountains. The reason for this is that GR is a purely classical model – that means it does not ( and cannot ) incorporate any quantum effects, which will become important on small scales.
Domain of Applicability. General Relativity is a purely classical model, so its domain of applicability is restricted to those situations where quantum effects can be neglected. When quantum effects become important, GR ceases to be an accurate description.
Before we go any further, it is imperative to be absolutely clear on what space-time curvature is not. Firstly, as we have seen, it is not a mechanical distortion of any kind of substance with mechanical properties; the term “space-time fabric” is often used in pop-sci media, but it is not to be taken literally – GR as a model doesn’t postulate space-time to be a substance or fabric of any kind. As stated above, GR does not, in fact, have anything at all to say as to the microscopic nature of it; it models only its macroscopic geometry. Secondly, space-time is not this  :
When you Google the term “space-time” and go into the “Images” tab, you get hundreds upon hundreds of variations of this theme. It depicts a bowling ball in the centre of some kind of sheet, and a marble rolling around it. This is called the “rubber sheet analogy”, and has probably done more damage to the general public’s perception of General Relativity than every pseudoscientist in history has ever managed to do. That is because this analogy is highly misleading in several key aspects, one of which I have already mentioned – it envokes the idea that space-time must be some kind of “fabric” that is mechanically distorted. The problem is that people unfamiliar with the theory of relativity, upon being shown the above graphic, tend to take this analogy literally and form questions such as :
- Doesn’t gravity already have to be present outside the sheet, in order for the ball to make a dent in it ? What pulls the ball down ?
- If gravity is supposedly space-time curvature, what makes the marble move in the first place ?
- Can’t I make the ball heavy enough so that the fabric rips ?
- How can there be stable orbits – the marble always falls into the bowling ball !
- What happens when I turn the rubber sheet upside down ?
- If gravity is space-time curvature, what does space-time curve into ?
And so on. The analogy is flawed in many ways, but the four main problems I see with it, so far as lay people are concerned, are :
- It doesn’t explain gravity at all, since gravity already has to be present external to the sheet in order to make the objects “heavy”
- It gives the impression that objects are somehow outside/separate from space-time itself
- The analogy looks like space-time is somehow embedded into some other higher dimensional space, into which it curves
- It completely omits the time dimension, and therefore is unable to depict or explain any dynamics
It is of crucial importance for all readers of this article to understand that the rubber sheet analogy is precisely that – an analogy. It is not an actual model for gravity, and it is not equivalent to General Relativity itself. Space-time is not a sheet that gets distorted when an object is ( externally ) placed onto it. I cannot stress this enough – all analogies are flawed, because by definition they remove aspects of the full model in order to focus on just a specific concept. In this instance, the concept that is meant to be illustrated is only that a massive gravitating object ( the bowling ball ) changes the geometry of space-time in such a way that a test particle ( the marble ) will move towards it over time. That is all. It is an analogy to illustrate these simple points, but it is not meant to be taken literally as a model for gravity.
Rubber Sheet Analogy. This analogy illustrates only that massive gravitating bodies change the geometry of space-time such that test particles will approach the massive body over time. It is strictly an analogy, and not to be taken literally, or mistaken for a model for gravity.
Allow me to specifically address some of the possible misconceptions this analogy might evoke in unsuspecting people, because it is important to get the right picture here. First of all – and this has been said many times already – gravity isn’t a substance, so if you have a body such as a planet or a star, space-time is not somehow “displaced” in the same manner as a body submerged in water will displace same. Space-time exists equally in the exterior vacuum surrounding a body, as well as the interior of the body itself. Material bodies are space-time too. The only difference between these two cases is found in the geometry of space-time, as we will discover shortly.
Secondly, space-time curvature has nothing to do with anything being “bent” into some external, higher dimensional space. The relevant concept here is the difference between intrinsic curvature and extrinsic curvature. Extrinsic curvature arises when you embed a surface into a higher dimensional space, and define the curvature of that surface by fitting – e.g. – a circle onto a given point, like so  :
The extrinsic curvature of the surface at that point is then characterised by the radius of that circle. It is called “extrinsic”, because it is defined by a measurement taken outside of the surface itself, and hence it only works if the surface is embedded into some other, higher dimensional space. Intrinsic curvature on the other hand is defined by taking measurements on the surface itself, without making reference to anything external  :
Intrinsic curvature can be defined in several different ways, for example by taking measurements of the sum of angles in a triangle, via parallelism of curves, via areas of simple shapes, via the shortest connection between points, via transporting vectors along paths, and so on. All of these will differ from the Euclidean case if a surface has intrinsic curvature. It is important to remember that intrinsic and extrinsic curvatures are related, but they are not necessarily equal – for example, a cylinder has extrinsic curvature, but no intrinsic curvature, it can be cut lengthwise and unfolded into a flat sheet.
In General Relativity, the geometry of space-time is given solely in terms of intrinsic curvature – no embedding of space-time into higher dimensional spaces is assumed or required. We simply examine space-time by looking at how geodesics on it diverge or converge, which is a purely intrinsic measurement.
Space-Time. Space-time is not a substance, and is not being displaced if a body is placed in it; it exists both in exterior vacuum as well as in the interior of massive bodies. The geometry of space-time is purely intrinsic, so it is not embedded into any higher dimensional space into which it “curves”.
Lastly I have no option but to disappoint you – the unfortunate fact is that neither space-time itself nor its geometry can be accurately visualised or depicted. Indeed, if that was possible, we wouldn’t need to bother with such misleading things as the “rubber sheet analogy”. The problem is that space-time itself is a 4-dimensional construct, so any attempt to visualise or represent it graphically will necessarily mean that we have to discard at least some of its properties, e.g. by suppressing one or more dimensions. This always leads to a loss of information. Furthermore, space-time curvature is not just a single real number assigned to each event – in fact, in its most general manifestation it is a rank-4 tensor quantity ( called the Riemann curvature tensor ) with 20 functionally independent components, so to completely fix all aspects of space-time geometry, we would have to assign a set of 20 numbers to each and every event. Obviously, there is no simple way to visualise this – you can’t plot a function of 20 variables onto a sheet or a computer screen, and expect something sensible to appear. So unfortunately, there isn’t any way to visualise space-time and its geometry – the best we can do is pick out certain aspects of it, and visualise those. For example, the rubber sheet diagram is actually an embedding diagram for a specific space-time geometry called Schwarzschild space-time, and it visualises the relationship between distances measured very far from a central body as compared to the same measurement taken closer to the body. In this context and with this particular application in mind, it is a useful tool. We will discuss this in detail in a later article. For now, the best I can do is offer you a schematic depiction of what the purely spatial part ( no time ) of curvature would look like, in three dimensions  :
This is not to scale, nor is it numerically accurate in any way, it is just a schematic visualisation. Distances are stretched radially and elongated towards the mass, but increasingly squeezed horizontally as you go down lower. This gives you a rough idea of what spatial curvature is about. It is better than the rubber sheet, but of course it is missing the time-like direction, which we will see shortly to be of crucial importance.
So now that we know that gravity isn’t a force but a geometric property of space-time, it is of course obvious to ask just how exactly this geometry is related to sources of gravity, such as planets. To answer this, let us put our empirical findings from the earlier drawings of falling test particles into a table  :
We immediately discover a pattern here : outside the Earth, in vacuum, the deviations between geodesics – taken in the three spatial directions – sum up to exactly zero. This does not mean that there is no gravity and no geodesic deviation in vacuum – it means only that, if we take the contributions given by all spatial directions, their sum is zero, because geodesics converge in some directions, but diverge in others. On the other hand, the same is not true in the interior of the Earth – we can still examine the sum of all contributions from all spatial directions, but it is no longer zero. To understand what this actually means in physical terms, imagine you have not just one test particle somewhete in the vicinity of a massive body, but a small enough ball of many tightly packed test particles that don’t mutually interact ( like “ideal” coffee grounds ). The ball of test particles is initially at rest, and the volume of that ball is just that of a normal sphere. What happens when we release our ball ? Of course, the entire ball will start moving towards the central body as it ages into the future – it is made up of many individual test particles, each one of which follows a geodesic. As we have seen, these geodesics will diverge and converge, so our ball will overall be distorted in shape as it continues to fall. However, this is not just any distortion, but it is such that the contributions from all spatial directions add up to zero – geometrically this means that the shape of the ball will change, but its volume will remain the same at all times  ( plotted here as a disk ) :
This is always true so long as the test particles are in vacuum ( i.e. outside any massive bodies ), and this preservation of volumes is what characterises the geometry of space-time in vacuum.
The same is not true in the interior of Earth – a small ball of test particles falling there will experience a distortion in both shape and volume during free fall.
Space-Time Curvature. In vacuum and during free fall, small test bodies will be distorted in shape, but retain their volume at all times. In the interior of mass distributions, both shapes and volumes will ( in general ) be changed.
So, physically speaking, what is difference between a vacuum region, and a region that is not vacuum ? The difference is the presence of mass. More precisely, it is the presence of mass or energy – for example, we could have a region that is filled with an electromagnetic field; there is no mass there, but there most certainly is energy. Furthermore, we know that for cases with weak gravity and slow speeds, Newtonian gravity evidently provides a very good description; and in Newtonian gravity, the source of the field is not mass per se, but rather mass density :
wherein is the local mass density. We demand ( quite reasonably so ) that General Relativity should agree with Newtonian gravity in the weak field limit, so GR should also use densities instead of mass and energy itself. We therefore postulate that there should be a relationship between the local curvature of space-time, and the local density of mass and energy, in order to accurately model the effects of gravity.
Since we have already looked in detail at how geodesics deviate, we might as well use our findings as a measure of how space-time is curved – the mathematical object that encapsulates the spatial curvatures in the various directions is called the Einstein tensor, and it is denoted by . This should be proportional to the densities of mass and energy, which is given by another mathematical object, called the stress-energy-momentum tensor, denoted by . We hence propose the following very simple relationship :
wherein is a proportionality constant that fixes the units between the two sides of the equation, and ensures that, for weak fields, everything agrees with Newtonian gravity. Equation (2) is called the Einstein field equation.
Einstein Field Equation. The collection of local spatial curvatures taken in all directions, which determine how geodesics deviate, equals the local density of energy-momentum, up to a proportionality constant.
Space-time curvature is equal to the distribution of energy-momentum in space-time.
The Einstein Field Equation is a system of equations to determine the components of the metric.
I realise that this, as presented, is of course all very ad-hoc and hand-wavey; for the purpose of this article I am deliberately choosing to not spend any time on making this mathematically precise, as it is more important here to convey the general meaning, rather than the rigorous maths. You will for now just have to take my word for it that equation (2) is indeed a good and valid mathematical description of the ideas presented throughout this text. Suffice to say that this can be mathematically derived in various rigorous ways, which is something we will be doing in another blog post.
At first glance, the Einstein field equation looks deceptively simple; however, on closer inspection, the first thing we realise is that each of the indices of the tensors run from 0…3, so in actual fact we are dealing with a system of equations ( 16 of them ) here, one equation for each choice of . The other problem we find is that the object is just a collection of scalar values which – once we choose a time direction – characterise how the corresponding spatial dimensions are curved at a given point. It doesn’t directly give us any information by how much ( in physical terms ) geodesics will deviate, in the sense that we can’t just “read out” that information simply by looking at the components of this tensor. In order to retrieve that info, we need to dig deeper, and break down the Einstein tensor in terms of a more fundamental object, the metric tensor. The metric tensor determines how measurements taken at different events in space-time are related to one another, and hence allows us to calculuate just by how much geodesics will deviate in a given situation. The Einstein tensor is calculated from the metric tensor, so in actual fact, equation (2), once written in terms of the metric tensor, looks like this :
wherein the -terms are functions of the metric and its derivatives. This is a system of partial differential equations to determine the components of the metric tensor. Unfortunately, each equation is highly non-linear, and makes reference to all components of the metric tensor and its derivatives, so we are left with a colourful hodge-podge of the mathematical variety. The equation system (3) has no general solution – it is possible only to find special solutions for given scenarios, and a full classification of all possible special solutions is still an unsolved mathematical problem. Generally speaking, solving (3) in closed analytical form requires us to make as many simplifying assumptions as we can, so that we are left only with a small number of non-vanishing components in the various tensors. To date, quite a number of exact analytical solutions is known for different physical scenarios, and many more can be obtained numerically with the help of computers. Each solutions describes a specific space-time with its corresponding geometry, for a specific physical setup. For example, space-time around a single and isolated, spherically symmetric, non-rotating and uncharged, static and stationary object is described by the Schwarzschild metric, being a solution to (2). If the object is rotating, you get a different solution, the Kerr metric, and so on. We will survey the most important classes of solutions in a future article.
Before we return to the physical consequences of GR, there are a few caveats and pitfalls I need you to be aware of. Firstly, the Einstein field equation (2) – which we will henceforth abbreviate as “EFE” – is a tensor equation, and tensors are purely local mathematical objects. Each of the tensors is defined not for a larger region, but rather at each event in space-time. That means that (2) is a local equation – it connects the local geometry of space-time with local sources of gravity. As such, the Einstein equations do not “automatically” account for distant sources, such as bodies that are far away from the point in question, or gravitational radiation originating somewhere far away; if it is required to account for the presence of such distant sources, we need to do so manually, by imposing boundary and initial conditions. This is no problem, since the field equations are differential equations, so boundary conditions are required anyway to obtain a solution.
Locality. The EFE connects only local space-time geometry with local sources of gravity. The influence of distant sources has to be accounted for separately in the form of boundary conditions.
Next, you need to be very mindful of the physical meaning of the tensors that appear in the equation. The energy-momentum tensor is discussed in some detail in my article Special Relativity : The Rest Of The Story, so it is probably a good idea to have a read through this, if you haven’t already done so. If we are in a vacuum, i.e. outside a massive body, the energy-momentum tensor vanishes ( ), so in vacuum the EFE becomes simply
You would be forgiven to think that this means that space-time in vacuum is perfectly flat, but that is wrong. That wouldn’t make any physical sense, since even if you are somewhere in empty space a few thousand miles from Earth, you would still feel its gravitational influence. The meaning of (4) is rather that the sum of the curvatures in the three spatial directions averages out to zero; it does not mean that there is no curvature. This just corresponds to the fact that geodesics will diverge in some directions but converge in others, and on average that gives zero ( see the table of deviations given earlier on ). But pick only one specific direction in space, and the deviation will not be zero, so space-time is not flat.
Einstein Tensor. The Einstein tensor is best understood as the average of all curvatures in an infinitesimally small region around a point; its vanishing ( G = 0 ) does not indicate that space-time is flat, but rather that the various curvatures in different directions average out to zero, due to having opposite signs.
Thirdly, the above two points taken together hint at an important limitation of the EFE – on its own, it does not uniquely fix all aspects of local space-time curvature. All it does is wire up local sources of energy-momentum with a quantity that gives us average curvatures over various directions; it can be shown that there are aspects of local space-time geometry that are left undetermined, until such time when we impose additional boundary and initial conditions. Hence, the EFE is merely a constraint on what form the metric tensor can take, it does not uniquely determine it. Furthermore, while the EFE constraints local geometry, it has nothing whatsoever to say about the global topology of space-time. I won’t go into this here, but it won’t hurt for you to be aware of it.
Limitations. The EFE represents only a constraint on what form local space-time geometry can take, but on its own it does not uniquely determine all aspects of it. Furthermore, it does not constrain the global topology of space-time at all.
Let us now return to the more physical aspects of General Relativity, and we will start with the relationship between space-time curvature and acceleration. This is a source of much confusion, but does lead us to an important principle.
We have thus far extensively spoken about what happens in free fall; but now suppose Albert E. is an astronaut travelling inside a rocket ship at a constant acceleration, and very far away from any massive objects such as planets or stars. Inside the ship, he will be pushed to the floor, just as he would on the surface of the Earth, and his accelerometer will read a non-zero value  :
But is Albert E. in a curved space-time ? If so, where is the source of that curvature ? At first glance, cases (A) and (B) above don’t appear to be distinguishable – in both cases, a local accelerometer will read the same value g.
As it turns out, Albert E. is not in fact in a curved space-time, even though the local effects inside the rocket ship superficially mimick the effects one would feel under the influence of a gravitating body, such as a planet. I say “superficially”, because there is one crucial difference – there is no tidal gravity inside the rocket ship. What this means is that the downwards acceleration is perfectly uniform everywhere in the cabin – two test particles placed anywhere in the ship will fall vertically, but they will remain parallel and not deviate horizontally at all, no matter where in the cabin we perform the experiment. This is quite unlike the case of test particles falling towards a massive body – they fall towards the body, but their trajectories will cease to remain parallel in the process, unlike is the case in our rocket ship. Here in the ship, both magnitude and direction of the acceleration is the exact same at all points. This leads us to  :
The Weak Equivalence Principle (I). Uniform acceleration is equivalent to the presence of a uniform gravitational field. There is no tidal gravity in a uniform gravitational field, and space-time in such a field is locally completely flat.
Note that the emphasis is on the word uniform, because that is the characteristic that distinguishes flat space-time from curved space-time – the latter will lead to the presence of tidal gravity, whereas the former will not. Because I have not ( within this article ) given you the tools to mathematically distinguish flat from curved space-times in an unambiguous way, I won’t attempt to provide proof of the above. Suffice to say the proof is straightforward – if you have read my blog post Manifolds and Curvature, you will already know that flatness implies the vanishing of the Riemann tensor, which provides a simple way to proof this mathematically.
If you look back at what we have done so far in this article, you will notice that I keep talking about “test particles”, but without ever making reference to the nature or properties of such particles. In fact, the very definition of a “geodesic” is simply that an accelerometer moving along such a world line reads zero at all times – it does not matter how that accelerometer is constructed, what mass it has, or what shape. This leads us to another formulation of the same principle :
The Weak Equivalence Principle (II). Geodesics of test particles in free fall depend only on initial conditions ( initial position and velocity ), but do not depend in any way on the structure, mass, or composition of the test particles.
This is demonstrated nicely in a famous experiment – if you drop a lead ball and a feather in a vacuum tube ( no air friction ), they will hit the ground at the same time, because they fall in the same manner.
In the above experiment with Albert E.’s rocket ship, we have eliminated tidal gravity by going far away from all sources such as planets, stars etc etc. However, there is another way to eliminate tidal gravity – just make the laboratory / rocket ship / frame small enough. So long as it is small enough, tidal gravity will play no role, because it is not a local effect and hence manifests only across relatively larger regions. Therefore, if we demand locality, i.e. so long as we ensure that our test laboratory is small enough so that tidal gravity can be neglected, then it does not matter at all precisely where we perform our experiments; it can be anywhere in space-time. In fact, it does not even matter if those experiments have anything to do with acceleration or not, so long as magnitude and direction of any acceleration is the same at all points within the lab. This gives us
The Strong Equivalence Principle (I). The outcome of any purely local experiment performed in a freely falling test laboratory does not in any way depend on the location in space-time, or the velocity of the test lab.
This is always true, so long as strict locality is maintained, i.e. so long as there is no tidal gravity, and no other external influences. Because that implies that there is no space-time curvature in our lab, we can also formulate the above in a different way :
The Strong Equivalence Principle (II). Every small enough region of space-time ( i.e. a small local patch over a short period of time ) is locally the flat Minkowski space-time of Special Relativity.
This sounds complicated, but corresponds to a very simple observation – locally, in a small area, the surface of the Earth looks perfectly flat, even though globally it is of course a sphere. Zoom in far enough into a small patch, and the overall curvature becomes negligible. The same is true for space-time too; every small enough reference frame is approximately flat, and can be treated with the simple laws of Special Relativity  :
Now let us think about some of the physical consequences of the equivalence principle in its various forms. We know now that, if we fire the thrusters of our rocket ship and “hover” stationary at some point within a gravitational field, then the effect within the rocket will be the same as if the rocket was accelerating. What does this imply for the world line of our intrepid astronaut ? We know from the previous Special Relativity articles that the longest possible world line in space-time is an inertial one, and if we add acceleration, world lines become relatively shorter. In other words, if we hover stationary at a point within a gravitational field, we experience less time as compared to some reference clock very far away. This phenomenon is called
Gravitational Time Dilation. A clock under the influence of a gravity will accumulate less proper time than a reference clock somewhere far away. The stronger the gravity ( = the higher the acceleration required to remain stationary ), the more time dilation.
For a case where the gravitational field is roughly spherically symmetric ( e.g. the Earth ), this quite simple means that the closer you are to the planet, the less proper time you accumulate on your clock  :
At first glance gravitational time dilation seems to be a very different effect than the kinematic time dilation of Special Relativity – the former is due to differences in position within a gravitational field ( = differences in gravitational potential ), whereas the latter is due to relative motion. However, on closer inspection it is found that both have the same origin – being the fact that the two clocks do not share the same notion of simultaneity. Time is purely local.
But wait a minute – how does this fit in with our concept of space-time curvature ? After all, we have claimed that small reference frames with approximately uniform gravitational fields / acceleration are perfectly flat. So why are clocks gravitationally dilated at all ? Why do we say that time is local ? To understand this, you need to remember that time dilation is not something that “happens” to a clock ( as in – some kind of “change” to its mechanism etc ), but rather a relationship between clocks; we see time dilation only if we compare two clocks. The thing here is just this – while the small, local frame of each clock itself is approximately flat, the region of space-time in between the clocks most certainly is not. This is nicely illustrated with the analogy of the table mountain  :
The tops of the mesas are roughly flat; the desert on which they stand is also roughly flat; however, in between the flat mesa top and the flat desert is a region of a very steep gradient. The same is true for space-time – two stationary clocks at different places in a gravitational field are each in locally flat small patches, but in between them is a region of curved space-time. To be more precise, the relevant curvature in this instance is a curvature in time, not a curvature in space. Even if the gravitational field was perfectly homogeneous in the spatial directions, we would still get gravitational time dilation.
Curvature in Time. Gravitational time dilation is a manifestation of curvature in the time direction ( as opposed to the spatial directions ). It is not a local effect, but something that is noticeable only across an extended region. Ultimately, it is due to the fact that time is a local concept, so clocks at different places do not share the same notion of simultaneity.
This effect isn’t just a theoretical conjecture – it is an empirical finding. We can measure the effect – as in the graphic above – by placing sensitive atomic clocks at different heights above Earth, and we will find that the proper time they accumulate will differ. But there is actually an even simpler way to measure this effect; if time is dilated between two ends of a vertical distance, then the frequency ( cycles per unit time ) of a ray of light travelling along that distance should experience a shift. It should becomes lower ( redshift ) if travelling away from the central body, and higher ( blueshift ) when in free fall towards it. Given sensitive enough receivers, we can test this effect even across a relatively small distance, for example inside a tall building  :
This is called the Pound-Rebka experiment. Because the frequency shift of the light beam is a direct consequence of gravitational time dilation, the Pound-Rebka experiment is in essence a way to directly measure space-time curvature, or more specifically curvature in time.
Pound-Rebka Experiment. This experimental setup detects dilation between clocks at different places in a gravitational field, and therefore is a direct measurement of space-time curvature.
Of course, this experiment has great value in experimentally verifying predictions made by our theory of General Relativity. But, it also demonstrates a fundamental concept within the theory, which is of crucial importance to everything that happens in gravitational physics :
The Principle of Locality. In curved space-times, all concepts of space and time, and measurements pertaining to those notions, are purely local. Observers in different places may disagree on measurements of space and time, which are no longer universally applicable concepts.
The Pound-Rebka experiment is an example – observers at the top of the tower, and observers at the bottom of the tower, will disagree on the wavelength and frequency of the same beam of light. That is because they perform these measurements with their own methods to label units of time and space, and those concepts are not the same at the two different locations. Time and space are purely local notions, they have no universal or absolute meaning. It is crucially important to understand this, because if you don’t, then GR as a model will appear to be full of strange, apparent paradoxes. Such as this one – because the speed of light is the same throughout, yet the two observers measure a different wave length, they must necessarily disagree on the energy in the beam of light. On the other hand though, every point along the light beam is locally just flat Minkowski space-time, so energy-momentum is conserved everywhere along the beam. So how can they disagree ?! What happened to the energy ?
Conservation of Energy-Momentum. Energy-momentum is locally conserved at each event in space-time, but there is no conservation law that applies across extended regions of curved space-time. There are no sources or sinks of energy-momentum anywhere, yet it is still not conserved globally.
Space-time curvature is a barrier to the existence of a global ( i.e. across larger regions ) conservation law for energy-momentum. Such a law is not violated, it just simply does not exist in the first place. That is because energy is intrinsically linked with the notion of time via Noether’s theorem, and the notion of time is not the same globally in a curved space-time, so no conservation law can be meaningfully defined ( not quite true, but good enough for our purposes here ). We will, in the not too distant future, dedicate an entire blog post the the topic of energy in curved space-times.
We have learned earlier that objects in free fall trace out geodesics in space-time, such that accelerometers always read zero everywhere. But what about the case of us standing on the surface of the Earth ? How does space-time curvature explain our everyday experience of gravity, which clearly hasn’t got anything to do with free fall and geodesics ? To understand this, we evoke a principle we have encountered earlier, the principle of extremal ageing. This just says that physical systems have a tendency to trace out world lines that maximise their proper time, i.e. they have a tendency to “choose” the longest possible world line they can. This is a manifestation of an even more fundamental principle, called the “principle of least action” – I won’t talk about this here, as it would lead us too far away from the subject matter of this article. Suffice to say that these are fundamental principles that apply to the evolution of all physical systems, regardless of whether these are gravitational or not, and classical or not. It’s a general principle of nature – in fact it is so fundamental that the Einstein field equations ( and hence GR ) can be directly and mathematically derived from it !
Put the pieces of the puzzle together : the principle of extremal ageing means that there is a tendency for you to get to a state where you experience the longest possible proper time on your clock, as compared to a reference clock somewhere far away; that means that there is a tendency for you to move towards regions of greater time dilation as you age into the future, with the most time dilation being experienced at the centre of the Earth. Of course, you can’t get there, because Earth’s surface is in the way – it counteracts your tendency to free fall, and that is just precisely the force you feel “resisting you” as you stand on the surface. In other words :
Everyday Gravity. Our everyday experience of gravity as weight ( downward force ) is due to curvature in the time direction. Tidal forces during free fall are due to curvature in the spatial directions.
While these two aspects of overall space-time curvature follow the same principles, for a simply body such as a planet, it is largely curvature in time that is responsible for the fact that test particles are gravitationally “attracted” to it, with the underlying mechanism being the principle of extremal ageing. Curvature in time is “more pronounced” than curvature in space by several orders of magnitude; these two aspects are also of opposite sign  :
Time dilates in the sense that ticks on a clock near a massive body are more “spaced out” as compared to a reference clock far away. Likewise, space is contracted in the sense that any given distance near a massive body corresponds to less distance on a ruler somewhere far away. What you consider “contracted” and “dilated” is of course largely a matter of convention, so long as you are being consistent about the usage of these terms. If you want to get a more interactive feel for how proper time behaves under the influence of a gravitating body, I recommend this online applet; feel free to play around with it a bit.
Given all of this, we are now in a position to reinterpret the famous “Newton’s Apple” in terms of space-time curvature, rather than the old concept of force  :
This video provides one of the best visualisations of curvature that I have seen so far – no “rubber sheet” here, just a straight plot of what happens to the apple over time. It captures the essence of General Relativity very nicely, and, in one form or another, summarises most of what we have talked about. If you replace the apple with any arbitrary test particle, you will get a good understand of how and why bodies move in curved space-times. Now, you may have noticed three things while reading this article :
- Almost nowhere at all have I used the term “force” – this is of course quite deliberate and on purpose. General Relativity represents a paradigm shift away from the old Newtonian way of thinking about mechanics in terms of invisible forces acting instantaneously across arbitrarily large distances, towards an understanding of the workings of nature in terms of geometry. If you really think about it, the Newtonian concept of force is really rather ludicrous – an invisible, intractable action over distance, which moves at infinite speed and can be arbitrarily large. In GR we go away from this notion, and consider everything in terms of how events are related to one another.
- I haven’t made reference to any of the numerous specific physical effects of General Relativity, such as frame dragging, event horizons, geodetic precession etc etc. This is because such effects depend on specific geometries, and it is not the purpose of this article to discuss such specifics. What I was trying to do here was present general principles which hold true everywhere in GR, not just in particular space-times. As such, everything we have talked about is applicable regardless of what specific geometry we are looking at.
- I have largely avoided mathematics and proofs. Again, this is on purpose. While GR is undoubtedly a mathematical model, I am a firm believer in that its fundamental principles as presented here ( if not the finer details ) can be understood by anyone, without necessarily having to be proficient in the ( sometimes seemingly arcane ) art of tensor calculus. This does of course not mean that mathematics can be dispensed with – a deeper understanding of GR is possible only when one understands the mathematics behind it. Nonetheless, this is one of those cases where a top-down approach seems in order – understand the reasoning behind the model first, then drill down to the mathematical details. There really is no point in starting to reel off pages of tensor manipulation, if the student does not have an understanding why we bother doing it in the first place.
General Relativity is such a vast subject that you could easily pick out any paragraph in this text, and turn it into an entire textbook. I have not done justice to the richness of the model by any stretch of the imagination, but in my opinion, trying to cram it all into a single blog post simply won’t work. I will therefore cut the presentation off at this point, because I feel that I have at least presented ( if not necessarily motivated or proven ) the main, overarching principles of GR. To summarise again, these principles are :
- Space-time and everything in it is a static model; real-world dynamics are examined by looking at the geometry of their ( static ) world lines
- All dynamics are driven by the fact that objects age into the future ( they can’t “stand still in time” )
- The natural state of motion of all objects is free fall – no proper acceleration is recorded here
- The Principle of Extremal Ageing : trajectories traced out in free fall are such that total proper time recorded is maximised.
- The relationships between events in the presence of gravitating bodies is the same as the relationship between points on a curved manifold. Space-time is not flat, but curved
- Gravity is geodesic deviation in a curved space-time
- General Relativity is a purely classical model, and cannot describe situations where quantum effects become important
- Space-time curvature is intrinsic – no embedding into a higher dimensional space is required or assumed
- Space-time is not a substance with mechanical properties
- The relationship between local sources of gravity and local geometry is given by the Einstein Field Equations
- Weak Equivalence Principle : locally in a small region, uniform acceleration is equivalent to the presence of a uniform gravitational field. The geodesics of test bodies do not depend on their structure or composition
- Strong Equivalence Principle : every small enough local region is Minkowskian. That means that the outcome of experiments performed there does not depend on position or velocity.
- Everyday gravity is largely a result of curvature in time
- Space-time curvature is not just an abstract concept, but a real phenomenon that can be directly detected and measured
- Principle of Locality : in General Relativity, there are no universally applicable concepts of space and time; these are purely local notions
- The law of energy-momentum conservation is not defined for regions of curved space-time.
The one thing I have not spoken about is the issue of symmetries, specifically the concept of diffeomorphism invariance. We will look at the mathematical formulation of General Relativity in the next blog post, which will be the right time to properly motivate diffeomorphism invariance. The next article will therefore be a rather more technical exposition, but like I said, this is equally important as the proper understanding of the fundamental principles. Truth be told, I have gone as far I can without making reference to the mathematics.
If someone was to ask me to summarise General Relativity in a single statement, then it would probably have to be this :
General Relativity. There are no universally valid notions of space and time; all concepts of space and time are purely local, and hence events are related in non-trivial ways. Observers at different places and times do not share the same notion of simultaneity.
That is the plain text essence of GR, without mathematics; everything else pretty much follows from this by logical reasoning and deduction. We will, in the upcoming blog posts, look at some specific examples of space-time geometries, and examine exactly what kind of physical manifestations arise from space-time curvature.
Until next time – stay tuned !