In today’s post, we shall take a closer look at the principles which underlie the theory of special relativity. I shall attempt to do this in a way so as to minimize the use of maths ( though there might be still be some ), in favor of an intuitive understanding of what the theory is about; as always, articles such as this one are not a substitute for a thorough study of an established textbook on the subject matter. At the same time, too many textbooks focus solely on performing mathematical calculations, but neglect to explain in detail just why we bother, and what those maths are really about. The purpose here is not to justify and motivate every single result and principle, but rather to present those principles in an understandable manner; the reader is urged to consult appropriate external resources for the deeper details.
In a nutshell, the theory of special relativity concerns itself with measurements taken at different times, different places, and different states of motion, and how those measurements are related to one another, in the absence of gravity. The inclusion of gravity will lead to a generalization of this to General Relativity, which we will discuss in another article. To be more specific, we are dealing with three types of measurements : measurements of time, as taken by clocks; measurements of space, as taken by rulers; and measurements of acceleration, as taken by accelerometers. As is the case with many other physical quantities too, these three fundamental concepts are defined via their measurements :
Time is what clocks measure. Space is what rulers measure. Acceleration is what accelerometers measure.
When we speak of clocks, rulers and accelerometers, we are referring to idealized instruments, in the sense that we do not presuppose any specific mechanism for them. For example, a “clock” can be a digital clock, a mechanical clock, an atomic clock etc etc, and the observations will always be the same. We just need to take care that our clock mechanism does not depend on any external influences – for example, a pendulum clock depends on external gravity, and is as such not a universally applicable way to measure time.
In order to conveniently compare measurements and analyze their relationships, we introduce the concept of space-time. Space-time is a mathematical model, which is quite simply the set of all events; it also contains information about how those events are related to one another. An event, in turn, is quite simply a specific point in space at a specific instant in time – in spite of the name ( which is convention ), it does not imply any form of dynamic, it is just a tuple of four numbers of the form (t,x,y,z), meaning a point at a given location and time. Obviously, what these numbers mean depends on where/when we choose the origin of our coordinate system to be – we cannot assign a “label” ( i.e. coordinates in the form (t,x,y,z) ) to an event unless we choose a coordinate system first. This choice of coordinate system defines a frame of reference – this simply means we decide on a specific way of how we wish to label events in space-time, in terms of coordinates.
The Theory of Relativity studies events in space-time, and how they are related to one another. It does so by analyzing the relationships between measurements taken with clocks, rulers and accelerometers at different times and places, and/or in different reference frames. Such frames determine how we choose to label events in space-time.
For our intents and purposes here, a frame of reference is synonymous with an observer – this is simply a method of performing measurements. Someone standing still on the surface of the earth is an observer ( with his own frame of reference ); someone traveling in a space-ship is an observer ( with his own frame ); someone falling into a black hole is an observer ( with his own frame ); and so on. These are all just different ways to take measurements of space, time, and acceleration.
Events in space-time are real, physical and immutable; everyone agrees on them. On the other hand, the labels we give those events – i.e. the choice of which coordinates we assign to them – is completely arbitrary, and depends on the observer.
What this means is that the laws of physics cannot depend on the specific labels we give events, since those labels are arbitrary – the laws of physics can only depend on relationships between physical events. One of the main goals of the theory of relativity is hence to find ways to formulate laws of physics in such a way that all observers agree on them – meaning we need to find ways to formulate laws so that their form does not depend on the system of coordinates we choose.
In order to do this, we need to take a closer look at the relationships between events. The easiest way to study those is to treat time and space on equal footing, and examine the resulting space-time geometrically. We treat the set of all events as a 4-dimensional construct, with three dimensions of space and one dimension of time; the easiest relationship between points in such a space is then their “distance”, i.e. their separation in space-time. It is important to understand that we are considering both space and time in this – the earth and the moon are separated in space, but likewise your house at 9am is separated from your house at 10am, even if it never physically moves with respect to the earth. Your house is stationary in space, but it ages forward in time, so “your house at 9am” is a distinct event from “your house at 10am” – and we can define the distance between these events in space-time.
Events can be separated in space, separated in time, or any combination of these.
There are infinitely many ways in which one could define the separation of events in space-time; in order to find the one that correctly describes the world around us, we need to resort to experiment and observation.
When you sit still on your comfortable sofa in your living room, and you turn on your laptop, it will function as expected – the screen comes on, the hard drive spins up etc etc. No surprises. Now suppose we consider an astronaut on the International Space Station ( ISS ), who has the exact same make and model of laptop as you do. The ISS moves in its orbit at a uniform speed of just over 17,000 miles per hour with respect to you in your living room. What happens when the astronaut turns on his laptop ? You intuitively know the answer – nothing special happens. The screen comes on and the hard drive spins up, exactly as it does in your living room. A laptop is just an engineering application of electricity, which in turn is just an application of the laws of electromagnetism and quantum mechanics. Let’s push this further, and examine what happens when we take the electrons moving in the circuits of our laptop, and make them move at higher and higher – but still uniform – speeds, for example in a particle accelerator. What we will find is that the quantum mechanical properties of our electrons, as well as the physics of how they interact, always remain the same, no matter how fast they move ! What this means is that the laws of physics remain the same, regardless of relative motion, so long as that motion is uniform and does not involve acceleration at the time when we examine the system. A frame of reference which is free of acceleration is called an inertial frame – an accelerometer placed into an inertial frame will read exactly zero.
The laws of physics are the same in all inertial frames.
Just to be clear on this again – this is not just a theoretical conjecture, but an empirical finding about the universe. The laws of physics are not in any way influenced by relative inertial motion; in particular, this applies to the laws of electrodynamics too. Since Maxwell’s equations are the same in all inertial frames, then so is the speed of light, since it follows directly from those equations. The very first consequence of this is that motion itself cannot be an intrinsic property of a reference frame – when you are weightless inside a windowless black box, and your accelerometer reads zero, you could be moving at the speed of the ISS ( 17,000 miles per hour ), or you could equally well be perfectly stationary. There is no way for you to tell the difference ( since there is none ! ), unless you open the box and look outside. In fact, it does not even make any sense to ask the question of what your “speed” is, unless you have an outside reference point to evaluate this against.
Motion is not an absolute, intrinsic property – it can only be meaningfully defined as a relationship between two frames.
So now we know that the laws of physics are the same in all inertial frames, they are invariant under the action of us placing our system in consideration into another inertial frame that is moving relative to us. What does this imply for how we define the separation of events in space-time ?
At this point, mathematics would really be needed to demonstrate how the conclusion follows from the premise; however, I have decided to refrain from listing equations, and will simply state the result :
The geometry of space-time is not Euclidean, but Lorentzian in nature.
Recall from your school days how – in a simple, 3-dimensional Euclidean space – you calculate the distance between points. In essence, you use the Pythagorean theorem to arrive at an expression of the form
It is tempting to extrapolate this to 4-dimensional space-time, however, it turns out that if we do that, we arrive at a situation where the laws of physics cannot remain the same if we change inertial reference frames ( mathematical proof omitted here ). To fix this, we need to give the “time” part of the separation between events an opposite sign as compared to the space part :
Or alternatively we could put all minuses in front of the space coordinates, and a plus in front of time – the result would be the same, this is just a matter of convention, and both ways are acceptable.
We now employ the tools of calculus, fix the dimensions, and let the coordinate differences become arbitrarily small; we can then rewrite (2) as
This is called the line element, and it allows us to define the distance/separation between events in space-time. In essence, it defines the separation between two infinitesimally close points on our 4-dimensional manifold; from this, we can calculate the length of curves, the areas and volumes of extended regions etc etc, simply by performing an appropriate integration of the line element. Generally speaking, it allows us to define arbitrary measurements in space-time. Take careful note of the constant c in front of dt – this constant has dimensions of m/s, and we need this to fix the dimensions in the expression – time is measured in seconds, and space is measured in meters, so in order to stay dimensionally consistent in the line element, we need to convert the dimension of either one of them, so that overall everything works out in the expression. If we put the c in front of dt, we get a separation between events that is measured in meters. We will find later that c is the called the “speed of light”, but it is crucially important to understand the true meaning of the constant, and look beyond the rather unfortunate nomenclature :
The “speed of light” constant c is a conversion factor between space and time. This coincides with the speed at which electromagnetic waves propagate.
So what does it mean when we say that the geometry of space-time is Lorentzian, i.e. what is the meaning of the space and time parts of the line elements having opposite sign ? To understand this, we flip the signs around a bit ( remember they just have to be opposite ) and write :
Now suppose we pick two events in your life that take place inside your house – call them event A and event B. You can travel between these events in various ways; for example, you could just stand still in your house, and travel between these events only through time. This is an inertial frame, since there is no acceleration involved. In that case, the spatial part of the line element is zero, and you are just left with
But you could also decide to continue your normal life; you keep going to work, walk around, travel etc etc. In addition to the difference in time, you have now also travelled through space, to arrive at the same event. So you now need the full line element again, which is (4). This motion between events A and B is now no longer purely inertial, since in order to return to your house, you had to change direction at some point, which would have caused your accelerometer to show a non-zero reading. Now compare the two trajectories between the same two events A and B – in the first case, you stood still and traveled only through time in a purely inertial frame; the distance along that trajectory is calculated with (5). In the second case you traveled through both time and space in a frame that is not purely inertial; the distance along that trajectory is calculated using (4). You will immediately see that – quite counterintuitively – the trajectory that spanned both time and space was shorter ! This allows us to write another fundamental principle, which also happens to be the distinguishing property of a Lorentzian ( as opposed to Euclidean ) geometry of space-time :
Inertial observers trace out straight lines between events in space-time; geometrically, such straight lines are the longest possible connection between events. Non-inertial observers trace out curves in space-time which are not straight; such curves connecting two events are always shorter than a straight line connection.
Inertial observers record the longest accumulated time between given events.
This seems very counterintuitive at first glance, but it is born out by empirical observation. If we replace our person in the house with a clock, and repeat the experiment, we find that the clock that remains inertial is accumulating the most time between events A and B. A reference clock experiencing acceleration at any point on its journey between these same two events will accumulate less time. This phenomenon is called proper time dilation. We will henceforth call the trajectory of an observer or measurement device its world line in space-time; a world line is just the spatial position of an observer plotted at every instant in time. We call the length of a world line between two events proper time :
The geometric length of a world line in space-time connecting two events is called proper time, and it is equal to the total amount of time a clock would physically record when it travels between those events. This forms the connection between abstract geometry, and real-world physical measurements.
In the above example, time dilation arises because one of the two world lines fails to be inertial. However, a similar concept can also occur between purely inertial frames. To understand why and how, consider a different experiment. Suppose we have a very fast moving, but very short-lived particle – such as a muon for example – being generated just above earth, and traveling towards a detector at the surface. Such processes do occur when high-energy cosmic radiation interacts with particles and radiation fields close to Earth. An observer stationed at the detector on the surface knows the average life time of a muon, and he knows in what region ( i.e. at what height ) such particles are produced, and at what uniform velocities they travel. From that, he can naively and easily calculate that no muon should ever reach the detector, since they all decay long before getting to the surface of earth ( simply put, they don’t travel fast enough to reach the surface within their limited life time ). We also know from previous experimentation that the laws of physics are the same for both observer and muon, since both can be approximated as inertial frames.
Now imagine the surprise on the observer’s face when he starts to detect plenty of cosmic muons in his lab ! How is that possible ? Both are inertial, and both are subject to the same laws ! So the observer double checks the variables – the muon does indeed move at the speed he know they would, and they are indeed produced at the height he knew they would, and they are indeed inertially moving. And yet they arrive. The only possible explanation – since the distance they travel is fixed and confirmed – is that somehow they experience less time, as compared to the observer on earth. This leads us to the following conclusion – not only is the relationship between inertial and non-inertial frames pretty non-Euclidean, but even the relationship between purely inertial frames moving relative to each other is non-trivial.
There are no absolute notions of space and time; measurements of distance and duration depend on the observer, and are not universally valid. Observers do not necessarily agree on such measurements.
If observers cannot agree on measurements of space and time when taken in isolation, it is immediately obvious that they also will not agree on notions of simultaneity, in the sense that readings on our Earth observer clock will not coincide with what he sees on the muon clock.
What is simultaneous in one frame may not be simultaneous from the point of view of another frame. This is called the “relativity of simultaneity”.
In the above experiment, we again replace the muon with a clock – the observer on Earth will see the clock coming at him at high speed to be going slower than his own reference clock. Because of this, the muon can reach him before it decays. The is the only possible explanation for this empirical finding, since we know the laws of physics to be the same in all inertial frames. The time recorded on a clock that is in a different frame than the observer himself is called coordinate time – in contrast to a reading on a clock that is in his own reference frame, which is the aforementioned proper time. Accordingly, the above phenomenon is called coordinate time dilation, which is a distinct phenomenon from proper time dilation.
Proper quantities – proper time, proper distance etc etc – are quantities physically measured in the same frame of reference as the observer. Coordinate quantities – coordinate time, coordinate distance etc etc – are what an observer calculates or observes in another ( possibly distant, or relatively moving ) frame. It is crucial not to confuse these.
But wait – what about the frame of the muon itself ? What does the muon see ? In that frame, the same laws of physics apply, so the same physical outcome must hold, namely that the detector is reached. The clock which measures the life time of the muon is now in the same frame as the inertially moving muon itself, so there cannot be any form of time dilation; the only way for the muon to reach the detector is therefore for the distance it needs to travel to be somehow shorter than in the frame of the earth-bound laboratory. This phenomenon is called length contraction, and it is the counterpart of time dilation.
Where one inertial observer sees time dilation, the other observer sees length contraction. These are two facets of the same physical phenomenon. Observers can therefore disagree on measurements of space and time, but they will never disagree on the final outcome of physical experiments ( which would create paradoxes ). In other words, observers may disagree on temporal and spatial separations, but they will always agree on the combined space-time interval between events.
The space-time interval is just the overall separation between events, as defined by the line element (4). If we have two fixed events in space-time, all observers must agree on this interval; on the other hand, they will disagree on distances in space and time, as demonstrated by the muon experiment. So what happens ? Well, if you look at the line element (4), you will notice that it is made up of two parts – a “time” part, and a “space” part. The difference between the two cannot change, so all that can happen is that contributions by time are shifted into the space part, and vice versa. And that is precisely the definition of “relative motion” :
The relationship between two frames in relative inertial motion with respect to one another is such that space is traded for time, and vice versa. Motion is a geometric relationship in space-time, not an intrinsic property of the frames themselves.
When you stand still, you age into the future, so you travel only in time; something moving relative to you travels both in time and space, but because you see the co-moving clock to be dilated, less time is “used up” to traverse more space. So time is traded for space, and the ratio between them is just relative speed. The limiting case would be something that travels only in space, but not in time – that case can never be realized though, because the degree by which the trade-off happens is not linear. Remember that our space-time has a Lorentzian geometry, not a Euclidean one; upon closer investigation it turns out this translates over into how changes in the distribution between the time and space components of the line element happen. The exact relationship can be written as an angle, like so :
What this means is that, in inertial motion between frames, time is “rotated” into space, and vice versa, and the rotation follows a law that is non-Euclidean. If you plot the above relationship, you will see that v/c can never be equal to one, because that would require an infinitely large rotation angle. Physically, that means that you cannot ever accelerate anything to the speed of light; it is simply not possible, on geometric grounds !
Going from one inertial frame into another inertial frame in relative motion, is equivalent to a hyperbolic rotation in space-time; inertial frames are related by hyperbolic rotations, so relative speed can be expressed as an angle, called rapidity. This transformation by rotating a coordinate system is called Lorentz transformation; this is how inertial frames are related in space-time. This transformation is invertible, so you can return to the original frame of reference by a second rotation about the same angle. That is why inertial frames are said to be symmetric.
The fact that the rotation follows the laws of Lorentzian geometry places stringent limits on what we can physically do in space-time; for one thing, there is no way to accelerate anything to or above the speed of light, so any dynamics originating at an event in space-time can propagate at most in a region bounded by this condition :
The above diagram is called an event’s light cone – it depicts space plotted against time ( not to scale ), and is named so because the surface of the cone represents the region where light waves would propagate. There is a future light cone and a past light cone, with the reference event itself in the middle. Only other events located inside or on the past light cone can influence our event, and likewise, our event can only influence other events that are located inside or on its own light cone. One can associate a light cone with each event in space-time, and this imposes a causal structure – only events that lie within a common light cone can be causally connected. For example, it is no problem for a rocket to travel from Earth to the moon within – say – six months. So, the events “earth right now” and “moon in six months Earth time” are causally connected. On the other hand, “earth right now” and “Andromeda galaxy in six months Earth time” are not causally connected, because nothing can travel that distance in six months, not even light. The event “Andromeda galaxy in six months Earth time” is outside Earth’s light cone.
So this is Special Relativity in a nutshell. It is a theory of measurements taken at and between events in space-time, and how those are related. Everything is described geometrically – even ( inertial ) motion itself, which now becomes nothing but a simple rotation in space-time ( a Lorentz transformation ). Let us summarize the main points :
- There are no absolute notions of space and time; measurements of rulers and clocks depend on the observer
- There are proper quantities and coordinate quantities – the former happen in the same frame as the observer, the latter in remote frames
- Inertial frames are related via hyperbolic rotations, so nothing can ever reach the speed of light, since the rotation angle would have to be infinitely large
- Motion is not an intrinsic property, but a relationship between frames in space-time
- All inertial frames are subject to the same laws of physics
- All observers agree on the separation between events in space-time, but not on measurements of space and time taken in isolation. This means that the line element is invariant under Lorentz transformations, i.e. it is the same in all frames.
A space-time with a line element of the form (4) is called Minkowski space-time; it has hyperbolic geometry, and it is completely flat. Much of what we have seen in this article is valid only for Minkowski space-time, and the one crucial thing we have not discussed at all is gravity. The inclusion of gravity into the theory of special relativity leads us to General Relativity, which we will look at in a future article.
The above is by no means an exhaustive treatment of Special Relativity. I have deliberately omitted a number of concepts, and kept gave some others only a cursory mention; and I have chosen not to go too deep into the mathematics of how things are quantified in this model. The idea here was to provide a “first primer”, and present the underlying ideas and principles, and not to get lost in pages after pages of minutely worked-out Lorentz transformations. I would like the reader to take this article as a starting point, and go on from here by consulting appropriate textbooks to learn all the fine details. In my mind, it is essential to first understand the motivations and overarching concepts, and then drilling down into the details.
Before I close, let me give you a word of warning :
Space-time is a mathematical concept relating events to one another; it is most emphatically not some kind of “fabric” with mechanical properties, in spite of what some people, and even some pop-sci sources, may claim. It is a model which we find to be in excellent agreement with real-world observations in situations where gravity can be neglected – not more and not less.
Keep this in mind, when, at a later stage, we go further on into General Relativity.
Follow-on article : Special Relativity : The Rest Of The Story