Space Time Geometry

[Vikki Nagindas]

Inphysics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects, such as how different observers perceive where and when events occur.

Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe). However, space and time took on new meanings with the Lorentz transformation and special theory of relativity.

In 1908, Hermann Minkowski presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. This interpretation proved vital to the general theory of relativity, wherein spacetime is curved by mass and energy.

Non-relativistic classical mechanics treats time as a universal quantity of measurement that is uniform throughout, is separate from space, and is agreed on by all observers. Classical mechanics assumes that time has a constant rate of passage, independent of the observer's state of motion, or anything external.[1] It assumes that space is Euclidean: it assumes that space follows the geometry of common sense.[2]

Unlike the analogies used in popular writings to explain events, such as firecrackers or sparks, mathematical events have zero duration and represent a single point in spacetime.[5] Although it is possible to be in motion relative to the popping of a firecracker or a spark, it is not possible for an observer to be in motion relative to an event.

In special relativity, an observer will, in most cases, mean a frame of reference from which a set of objects or events is being measured. This usage differs significantly from the ordinary English meaning of the term. Reference frames are inherently nonlocal constructs, and according to this usage of the term, it does not make sense to speak of an observer as having a location.[9]

Physicists distinguish between what one measures or observes, after one has factored out signal propagation delays, versus what one visually sees without such corrections. Failing to understand the difference between what one measures and what one sees is the source of much confusion among students of relativity.[10]

By the mid-1800s, various experiments such as the observation of the Arago spot and differential measurements of the speed of light in air versus water were considered to have proven the wave nature of light as opposed to a corpuscular theory.[11] Propagation of waves was then assumed to require the existence of a waving medium; in the case of light waves, this was considered to be a hypothetical luminiferous aether.[note 1] The various attempts to establish the properties of this hypothetical medium yielded contradictory results. For example, the Fizeau experiment of 1851, conducted by French physicist Hippolyte Fizeau, demonstrated that the speed of light in flowing water was less than the sum of the speed of light in air plus the speed of the water by an amount dependent on the water's index of refraction.[12]

In 1905, Albert Einstein analyzed special relativity in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics. His results were mathematically equivalent to those of Lorentz and Poincar. He obtained them by recognizing that the entire theory can be built upon two postulates: the principle of relativity and the principle of the constancy of light speed. His work was filled with vivid imagery involving the exchange of light signals between clocks in motion, careful measurements of the lengths of moving rods, and other such examples.[23][note 4]

When Einstein published in 1905, another of his competitors, his former mathematics professor Hermann Minkowski, had also arrived at most of the basic elements of special relativity. Max Born recounted a meeting he had made with Minkowski, seeking to be Minkowski's student/collaborator:[25]

I went to Cologne, met Minkowski and heard his celebrated lecture 'Space and Time' delivered on 2 September 1908. [...] He told me later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other was pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendor. He never made a priority claim and always gave Einstein his full share in the great discovery.

Minkowski had been concerned with the state of electrodynamics after Michelson's disruptive experiments at least since the summer of 1905, when Minkowski and David Hilbert led an advanced seminar attended by notable physicists of the time to study the papers of Lorentz, Poincar et al. Minkowski saw Einstein's work as an extension of Lorentz's, and was most directly influenced by Poincar.[26]

On 5 November 1907 (a little more than a year before his death), Minkowski introduced his geometric interpretation of spacetime in a lecture to the Gttingen Mathematical society with the title, The Relativity Principle (Das Relativittsprinzip).[note 5] On 21 September 1908, Minkowski presented his talk, Space and Time (Raum und Zeit),[27] to the German Society of Scientists and Physicians. The opening words of Space and Time include Minkowski's statement that "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." Space and Time included the first public presentation of spacetime diagrams (Fig. 1-4), and included a remarkable demonstration that the concept of the invariant interval (discussed below), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity.[note 6]

The spacetime concept and the Lorentz group are closely connected to certain types of sphere, hyperbolic, or conformal geometries and their transformation groups already developed in the 19th century, in which invariant intervals analogous to the spacetime interval are used.[note 7]

Although two viewers may measure the x, y, and z position of the two points using different coordinate systems, the distance between the points will be the same for both, assuming that they are measuring using the same units. The distance is "invariant".

In four-dimensional spacetime, the analog to distance is the interval. Although time comes in as a fourth dimension, it is treated differently than the spatial dimensions. Minkowski space hence differs in important respects from four-dimensional Euclidean space. The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two events (because of time dilation) or the distance between the two events (because of length contraction). Special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time. All observers who measure the time and distance between any two events will end up computing the same spacetime interval. Suppose an observer measures two events as being separated in time by Δ t \displaystyle \Delta t and a spatial distance Δ x . \displaystyle \Delta x. Then the squared spacetime interval ( Δ s ) 2 \displaystyle (\Delta s)^2 between the two events that are separated by a distance Δ x \displaystyle \Delta x in space and by Δ c t = c Δ t \displaystyle \Delta ct=c\Delta t in the c t \displaystyle ct -coordinate is:[32]

The constant c , \displaystyle c, the speed of light, converts time t \displaystyle t units (like seconds) into space units (like meters). The squared interval Δ s 2 \displaystyle \Delta s^2 is a measure of separation between events A and B that are time separated and in addition space separated either because there are two separate objects undergoing events, or because a single object in space is moving inertially between its events. The separation interval is the difference between the square of the spatial distance separating event B from event A and the square of the spatial distance traveled by a light signal in that same time interval Δ t \displaystyle \Delta t . If the event separation is due to a light signal, then this difference vanishes and Δ s = 0 \displaystyle \Delta s=0 .

Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general, x \displaystyle x means Δ x \displaystyle \Delta x , etc. We are always concerned with differences of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values have no essential meaning.

The light cone has an essential role within the concept of causality. It is possible for a not-faster-than-light-speed signal to travel from the position and time of O to the position and time of D (Fig. 2-4). It is hence possible for event O to have a causal influence on event D. The future light cone contains all the events that could be causally influenced by O. Likewise, it is possible for a not-faster-than-light-speed signal to travel from the position and time of A, to the position and time of O. The past light cone contains all the events that could have a causal influence on O. In contrast, assuming that signals cannot travel faster than the speed of light, any event, like e.g. B or C, in the spacelike region (Elsewhere), cannot either affect event O, nor can they be affected by event O employing such signalling. Under this assumption any causal relationship between event O and any events in the spacelike region of a light cone is excluded.[35]

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