Spacetime

In physics, spacetime 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.

"space and time" and "time and space" redirect here. For other uses, see Space and Time (disambiguation), Timespace (disambiguation), and Spacetime (disambiguation).

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.

Fundamentals[edit]

Definitions[edit]

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]

In the context of special relativity, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer.^[3]^{: 214–217} General relativity provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field.

In ordinary space, a position is specified by three numbers, known as dimensions. In the Cartesian coordinate system, these are often called x, y and z. A point in spacetime is called an event, and requires four numbers to be specified: the three-dimensional location in space, plus the position in time (Fig. 1). An event is represented by a set of coordinates x, y, z and t.^[4] Spacetime is thus four-dimensional.

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.

The path of a particle through spacetime can be considered to be a sequence of events. The series of events can be linked together to form a curve that represents the particle's progress through spacetime. That path is called the particle's world line.^[6]^: 105

Mathematically, spacetime is a manifold, which is to say, it appears locally "flat" near each point in the same way that, at small enough scales, the surface of a globe appears to be flat.^[7] A scale factor, $c$ (conventionally called the speed-of-light) relates distances measured in space with distances measured in time. The magnitude of this scale factor (nearly 300,000 kilometres or 190,000 miles in space being equivalent to one second in time), along with the fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe that is noticeably different from what they might observe if the world were Euclidean. It was only with the advent of sensitive scientific measurements in the mid-1800s, such as the Fizeau experiment and the Michelson–Morley experiment, that puzzling discrepancies began to be noted between observation versus predictions based on the implicit assumption of Euclidean space.^[8]

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]

In Fig. 1-1, imagine that the frame under consideration is equipped with a dense lattice of clocks, synchronized within this reference frame, that extends indefinitely throughout the three dimensions of space. Any specific location within the lattice is not important. The latticework of clocks is used to determine the time and position of events taking place within the whole frame. The term observer refers to the whole ensemble of clocks associated with one inertial frame of reference.^[9]^: 17–22

In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event and its recording. A real observer, will see a delay between the emission of a signal and its detection due to the speed of light. To synchronize the clocks, in the data reduction following an experiment, the time when a signal is received will be corrected to reflect its actual time were it to have been recorded by an idealized lattice of clocks.^[9]^: 17–22

In many books on special relativity, especially older ones, the word "observer" is used in the more ordinary sense of the word. It is usually clear from context which meaning has been adopted.

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]

The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′.

Frame S′ moves in the x-direction of frame S with a constant velocity v as measured in frame S.

The origins of frames S and S′ are coincident when time t = 0 for frame S and t′ = 0 for frame S′.^: 107

[6]

If u′ and v are both very small compared with the speed of light, then the product vu′/c² becomes vanishingly small, and the overall result becomes indistinguishable from the Galilean formula (Newton's formula) for the addition of velocities: u = u′ + v. The Galilean formula is a special case of the relativistic formula applicable to low velocities.

If u′ is set equal to c, then the formula yields u = c regardless of the starting value of v. The velocity of light is the same for all observers regardless their motions relative to the emitting source.^: 49

[41]

Closure under linear combination: If A and B are 4-vectors, then $C=aA+aB$ is also a 4-vector.

Inner-product invariance: If A and B are 4-vectors, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a 3-vector. In the following, ${\vec {A}}$ and ${\vec {B}}$ are 3-vectors:
$A\cdot B\equiv$ $A_{0}B_{0}-A_{1}B_{1}-A_{2}B_{2}-A_{3}B_{3}\equiv$ $A_{0}B_{0}-{\vec {A}}\cdot {\vec {B}}$

The first crucial concept is coordinate independence: The laws of physics cannot depend on what coordinate system one uses. This is a major extension of the principle of relativity from the version used in special relativity, which states that the laws of physics must be the same for every observer moving in non-accelerated (inertial) reference frames. In general relativity, to use Einstein's own (translated) words, "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion."^: 113 This leads to an immediate issue: In accelerated frames, one feels forces that seemingly would enable one to assess one's state of acceleration in an absolute sense. Einstein resolved this problem through the principle of equivalence.^[56]^{: 137–149}

[55]

Technical topics[edit]

Is spacetime really curved?[edit]

In Poincaré's conventionalist views, the essential criteria according to which one should select a Euclidean versus non-Euclidean geometry would be economy and simplicity. A realist would say that Einstein discovered spacetime to be non-Euclidean. A conventionalist would say that Einstein merely found it more convenient to use non-Euclidean geometry. The conventionalist would maintain that Einstein's analysis said nothing about what the geometry of spacetime really is.^[83]

Such being said,

; Tipler, Frank J. (1986). The Anthropic Cosmological Principle (1st ed.). Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.

Barrow, John D.

and Ruth M. Williams (1992) Flat and curved space–times. Oxford University Press. ISBN 0-19-851164-7

George F. Ellis

Einstein, Albert, Minkowski, Hermann, and Weyl, Hermann (1952) The Principle of Relativity: A Collection of Original Memoirs. Dover.

Lorentz, H. A.

(1973) A Treatise on Time and Space. London: Methuen.

Lucas, John Randolph

(2004). The Road to Reality. Oxford: Oxford University Press. ISBN 0-679-45443-8. Chapters 17–18.

Penrose, Roger

Taylor, E. F.; (1992). Spacetime Physics, Second Edition. Internet Archive: W. H. Freeman. ISBN 0-7167-2327-1.

Wheeler, John A.

(1 December 2017). The Doom of Spacetime: Why It Must Dissolve Into More Fundamental Structures (Speech). The 2,384th Meeting Of The Society. Washington, D.C. Retrieved 16 July 2022.

Arkani-Hamed, Nima

Media related to Spacetime at Wikimedia Commons

13th edition Encyclopædia Britannica Historical: Albert Einstein's 1926 article