Special relativity / Elementary Tour part 5: Spacetime

“All the world’s a stage.” – that’s how we’re used to viewing space: As a stage on which objects are located and where the dramas of their movements and evolution take place.

In special relativity, as was mentioned briefly, simultaneity is relative, and so are time and space. Observers moving relative to one another come to different conclusions about which events happen simultaneously (“at the same time”). They agree only about what events there are, not about where or when these events take place. Spacetime, the totality of all events, is absolute. But there are different ways to slice that collection of events into snapshots of simultaneity. When viewed in succession, these snapshots show how, with time, changes take place in space. Different observers, looking at different successions of snapshots, come to different conclusions about which events are simultaneous. Spacetime is absolute, space and time are not.

Nevertheless, spacetime does retain what physicists call a causal structure, which is the same for all (inertial) observers. This structure determines which events can, in principle, be influenced by which other events, and also for which pairs of events mutual influence is impossible.

For every event A, happening at a definite location at a definite time, and every event B, also defined by location and time, we can ask: Could A, by whatever means, be the cause of B? We are able to answer this quite generally because of the role of light speed as the absolute upper limit for the transmission of energy, matter and information. Whatever influence A could have exerted to cause B, such influence could have travelled no faster than the speed of light. If B happens too soon, too far away from A, so that a light signal starting at A would reach the location of B only after the event B had already happened, then any putative influence of A on B would have had to have travelled faster than light. On the other hand, if such a light signal were to reach the location of B before the event B actually happened, then any supposed influence from A could reach B without having to break the cosmic speed limit. We still wouldn’t know what influence that might have been, or whether it could have been transmitted in any realistic way. But we do have a rough notion of causality: For every pair of events, we can tell whether an influence is possible in principle, simply by looking how long it would take a light signal from A to reach the location of B.

In relativity, the “region of influence” of an event is often represented graphically, in the following way:

spacetime diagram

Such a figure is called a spacetime diagram, and the vertical axis represents time (as measured by a specific observer), while the horizontal axis represents space. Light signals produced or received at the event A (itself located where the two axes intersect) and moving away into space are represented by the diagonal yellow lines. They delineate a region of spacetime which is called the light cone of the event A – with the additional two space dimensions included, this region would be the higher-dimensional analogue of a double cone.

In the upper part of the light cone, denoted as region I in the figure and called the “future light cone”, you can find all events on which event A could have an influence (in other words: a light signal or slower-than-light influence from A could reach such an event). In the lower part, the “past light cone”, region II, lie all events by which A could have been influenced (again in other words: from such events, a light signal or slower-than-light influence could have made it to the location of A in time). Outside the double cone, in region III, lie all the events which are causally separated from A. Those events can neither influence A nor be influenced by A.

Previous part: The speed of light

Next part: E = mc²

Tour index

Definitioner

time

It is a fact of life that not all events in our universe happen concurrently - instead, there is a certain order. Defining a time coordinate or defining time, the way physicists do it, is to define a prescription to associate with each event a number so as to reflect that order - if event B happens after event A, then the number associated with B should be larger than that associated with A. The first step of this definition is to construct a clock: Choose a simple process that repeats regularly. (What is "regular"? Luckily, in our universe, all elementary processes such as a swinging pendulum, the oscillations of atoms or of electronic circuits lead to the same concept of regularity.) As a second step, install a counter: A mechanism that, with every repetition of the chosen process, raises the count by one. With this definition, one can at least assign a time (the numerical value of the counter) to events happening at location of the clock. For events at different locations, an additional definition is necessary: One needs to define simultaneity. After all, the statement that some far-away event A happens at 12 o'clock is the same as saying that event A and "our clock counter shows 12:00:00" are simultaneous. The how and why of defining simultaneity - a centre-piece of Einstein's special theory of relativity - are described in the spotlight topic Defining "now". With all these preparations, physicists can, in principle, assign a time coordinate value ("a time") to any possible events, and describes how fast or how slow processes happen, compared to that time coordinate.

speed

An object's average speed is the distance it moves during a given period of time, divided by the length of the time interval. If you make the time interval infinitely small, the result is the object's speed at one particular moment in time. The notion of speed can be applied to waves in different ways; for instance, for a simple wave, the phase speed is the speed at which any given wave crest or wave propagates through space. See also the more general entry velocity.

speed of light

The speed at which light or, more generally, electromagnetic radiation propagates through space (especially: through empty space). Central quantity in special relativity: There, the constancy of the speed of light is a basic postulate: every observer (more precisely: every inertial observer) that measures the speed of light in vacuum obtains the same constant value, c=299,792,458 metres per second. Another important relativistic aspect of the speed of light is that it defines an absolute upper speed limit: In special relativity, nothing can move faster than light, and information or influence at most be transmitted at light-speed. In general relativity, the same law is in force locally: No object, no matter, no information can directly overtake or catch up with light (cf. causality). Basic information about the role of light speed in special relativity can be found in the chapter Special relativity of Elementary Einstein.

special relativity

Albert Einstein's theory of the fundamentals of space, time, and movement (but not gravity). For a brief introduction, check out the chapter Special relativity of Elementary Einstein. Selected aspects of special relativity are described in the category Special relativity of our Spotlights on relativity.

space

In a strict sense: Space as we know it from everyday life: the totality of all locations in which objects can sit, with three dimensions. In a more general sense used by mathematicians, all kinds of sets of points are spaces - a line for instance, which has but a single dimension, or a two-dimensional surface, but also higher-dimensional spaces. Also, in such more general spaces, geometry can be different from the standard Euclidean geometry taught in high schools - such spaces can be curved.

spacetime

Already in special relativity, observers in motion relative to each other will not, in general, agree as to whether two events happen simultaneously, or as to how great is the distance between two objects. They do, however, agree as to what events there are, although not to when and where they happen. This observer-independent totality of all events is called spacetime. How spacetime is split into space and time can differ from observer to observer. Every-day space has three dimensions. Adding time adds another dimension - spacetime has four dimensions, all in all. We are used to the idea of a point in space - an object that has only one location and is completely defined once its space coordinates are given. In spacetime, a spacetime point is an object defined completely once its space coordinates and its time coordinate are given - which makes a spacetime point nothing but an elementary event. The idea of spacetime is, in addition to its role in special relativity, a building block of general relativity. Analogous to how a plane is flat, but the surface of a sphere is curved, in general relativity, curved or distorted versions of the simple, flat spacetime of special relativity play a role. Spacetime curvature, in general relativity, is intimately connected with gravity. For an introduction to the basics of both theories of relativity, check out the chapters Special relativity and General relativity in Elementary Einstein. Sometimes, it can be helpful to view spacetime in analogy to ordinary space - such analogies are explored in the spotlight topics Time dilation on the road (for time dilation) and Twins on the road (for the twin effect).

observer

In the context of relativity, "observer" can mean two different things. Often, observer is synonymous with reference frame or (spacetime-)coordinate system: An observer in this sense is someone who assigns coordinates to everything that happens around him. In particular, all events are assigned space coordinate values and a time coordinate value. In the context of special relativity, it is often the case that when one talks about an observer, what is really meant is an inertial observer, corresponding to a special type of reference frame. On other occasions, the term is used in a more narrow sense - in those cases, an observer is someone sitting at a certain point in space and using the light signals reaching that location to construct an image of his surroundings. In the context of optical effects in relativity, for instance gravitational lensing, observer is usually meant in this way.

matter

In general relativity: All contents of spacetime that contribute to its curvature: particles, dust, gases, fluids, electromagnetic and other fields. In particle physics: All elementary particles with half-integer spin, such as electrons and quarks, as well as their composites such as protons and neutrons, in contrast with force particles.

light

Light in the strict sense of the word is electromagnetic radiation the human eye can detect, with wave-lengths between 400 and 700 nano metres. In relativity theory and in astronomy, the word is often used in a more general sense, encompassing all kinds of electromagnetic radiation. For instance, astronomers might talk about "infrared light" or "gamma light"; in this context, light in the stricter sense is referred to as "visible light". Within classical physics, the properties of light are governed by Maxwell's equations; in quantum physics, it turns out that light is a stream of energy packets called light quanta or photons. In the context of relativistic physics, light is of great interest, and for a number of reasons. First of all, the speed of light plays a central role in both special and general relativity. Also, there are a number of interesting effects in general relativity which are associated with the propagation of light, namely deflection, the Shapiro effect and the gravitational redshift.

light speed

See speed of light.

event

In the context of special or general relativity: Anything that is defined by a single point in time and a single point in space, i.e. something that happens at a definite time at a definite location. Synonym: spacetime point.

energy

Physical quantity with the special property that, in physical processes, energy is neither destroyed nor created, simply transformed from one form of energy to another. Some of the different forms of energy that are defined separately are kinetic energy, thermal energy and the energy carried by electromagnetic radiation. Processes that transform one form of energy into another take place in all machines we use in everyday life, from the engine of a subway train (electrical energy into kinetic energy of the train) to an electric blanket (electrical energy into thermal energy). One important consequence of special relativity is that energy and mass are completely equivalent - two different ways to define what is, on closer inspection, one and the same physical quantity. See the keyword equivalence between mass and energy.

causal

In the context of relativity: causality concerns the question of which events cause which other events (Latin causa, the reason, the cause) or, more generally, can influence them. In special relativity, nothing, no moving object, no information, no influence can move faster than light. Thus, in principle, an event can only influence another event if the hypothetical influence (such as a signal or a force) would not have to be transmitted faster than light. In other words, light propagation determines the causal structure of spacetime (cf. light-cone). Models and theories that take this structure into account are called causal - for example the relativistic quantum field theories. In general relativity, the cosmic speed limit, the speed of light is only defined locally: In a side-by-side race, no object, no influence can overtake a light signal. Also from this a causal structure can be derived and determines which events can influence which other events. As gravity deflects and delays light signals, the matter is more complicated in general relativity than in special relativity. Although this makes the analysis somewhat more complicated, the following still applies: the propagation of light determines the causal structure.

Special relativity / Elementary Tour part 5: Spacetime

Further Articles

Tag Cloud