Editing Spacetime (section)

=== Spacetime interval ===
{{See also|Causal structure}}
In three dimensions, the ''[[Euclidean distance|distance]]'' <math>\Delta{d}</math> between two points can be defined using the [[Pythagorean theorem]]:
: <math>(\Delta{d})^2 = (\Delta{x})^2 + (\Delta{y})^2 + (\Delta{z})^2</math>

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 special relativity, however, the distance between two points is no longer the same if measured by two different observers, when one of the observers is moving, because of [[Lorentz contraction]]. The situation is even more complicated if the two points are separated in time as well as in space. For example, if one observer sees two events occur at the same place, but at different times, a person moving with respect to the first observer will see the two events occurring at different places, because the moving point of view sees itself as stationary, and the position of the event as receding or approaching. Thus, a different measure must be used to measure the effective "distance" between two events.<ref name="Kogut_2001">{{cite book |last1=Kogut |first1=John B. |title=Introduction to Relativity |date=2001 |publisher=Harcourt/Academic Press |location=Massachusetts |isbn=0-12-417561-9}}</ref>{{rp|48–50;100–102}}

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 space|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 [[event (relativity)|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 <math>\Delta t</math> and a spatial distance <math>\Delta x.</math> Then the squared spacetime interval <math>(\Delta{s})^2</math> between the two events that are separated by a distance <math>\Delta{x}</math> in space and by <math>\Delta{ct}= c\Delta t</math> in the <math>ct</math>-coordinate is:<ref name="D'Inverno_1002">{{cite book |title=Introducing Einstein's Relativity: A Deeper Understanding |author1=Ray d'Inverno |author2=James Vickers |edition=illustrated |publisher=Oxford University Press |year=2022 |isbn=978-0-19-886202-4 |pages=26–28 |url=https://books.google.com/books?id=LGxvEAAAQBAJ}} [https://books.google.com/books?id=LGxvEAAAQBAJ&pg=PA27 Extract of page 27]</ref>
: <math>(\Delta s)^2 = (\Delta ct)^2 - (\Delta x)^2,</math>
or for three space dimensions,
: <math>(\Delta s)^2 = (\Delta ct)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2.</math>

The constant <math>c,</math> the speed of light, converts time <math>t</math> units (like seconds) into space units (like meters). The squared interval <math>\Delta s^2</math> 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 <math>\Delta t</math>. If the event separation is due to a light signal, then this difference vanishes and <math>\Delta s =0</math>.

When the event considered is infinitesimally close to each other, then we may write
: <math>ds^2 = c^2dt^2 - dx^2-dy^2-dz^2.</math>

In a different inertial frame, say with coordinates <math>(t',x',y',z')</math>, the spacetime interval <math>ds'</math> can be written in a same form as above. Because of the constancy of speed of light, the light events in all inertial frames belong to zero interval, <math>ds=ds'=0</math>. For any other infinitesimal event where <math>ds\neq 0</math>, one can prove that <math>ds^2=ds'^2</math>
which in turn upon integration leads to <math>s=s'</math>.<ref>Landau, L. D., and Lifshitz, E. M. (2013). The classical theory of fields (Vol. 2).</ref>{{rp|2}} The invariance of the spacetime interval between the same events for all inertial frames of reference is one of the fundamental results of special theory of relativity.

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, <math>x</math> means <math>\Delta{x}</math>, 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.

[[File:Spacetime Diagram of Two Photons and a Slower than Light Object.png|thumb|Figure 2–1. Spacetime diagram illustrating two photons, A and B, originating at the same event, and a slower-than-light-speed object, C]]

The equation above is similar to the Pythagorean theorem, except with a minus sign between the <math>(ct)^2</math> and the <math>x^2</math> terms. The spacetime interval is the quantity <math>s^2,</math> not <math>s</math> itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard <math>s^2</math> as a distinct symbol in itself, rather than the square of something.<ref name="Schutz" />{{rp|217}}
: '''Note:''' There are two sign conventions in use in the relativity literature:
:: <math>s^2 = (ct)^2 - x^2 - y^2 - z^2</math>
: and
:: <math>s^2 = -(ct)^2 + x^2 + y^2 + z^2</math>
: These sign conventions are associated with the [[metric signature]]s {{nowrap|(+−−−)}} and {{nowrap|(−+++).}} A minor variation is to place the time coordinate last rather than first. Both conventions are widely used within the field of study.<ref name="Carroll_2022">{{cite book |last1=Carroll |first1=Sean |title=The Biggest Ideas in the Universe |date=2022 |publisher=Penguin Random House LLC |location=New York |isbn=9780593186589 |pages=155–156}}</ref>
: In the following discussion, we use the first convention.

In general <math>s^2</math> can assume any real number value. If <math>s^2</math> is positive, the spacetime interval is referred to as '''timelike'''. Since spatial distance traversed by any massive object is always less than distance traveled by the light for the same time interval, positive intervals are always timelike. If <math>s^2</math> is negative, the spacetime interval is said to be '''spacelike'''. Spacetime intervals are equal to zero when <math>x = \pm ct.</math> In other words, the spacetime interval between two events on the world line of something moving at the speed of light is zero. Such an interval is termed '''lightlike''' or '''null'''. A photon arriving in our eye from a distant star will not have aged, despite having (from our perspective) spent years in its passage.<ref name="Kogut_2001"/>{{rp|48–50}}

A spacetime diagram is typically drawn with only a single space and a single time coordinate. Fig.&nbsp;2-1 presents a spacetime diagram illustrating the [[world lines]] (i.e. paths in spacetime) of two photons, A and B, originating from the same event and going in opposite directions. In addition, C illustrates the world line of a slower-than-light-speed object. The vertical time coordinate is scaled by <math>c</math> so that it has the same units (meters) as the horizontal space coordinate. Since photons travel at the speed of light, their world lines have a slope of ±1.<ref name="Kogut_2001"/>{{rp|23–25}} In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.

{{anchor|Reference frames}}