Physical definition

From the age of Newton up until Einstein's profound reinterpretation of the physical concepts associated with time and space, time was considered to be "absolute" and to flow "equably" (to use the words of Newton) for all observers.^[34] The science of classical mechanics is based on this Newtonian idea of time.

Einstein, in his special theory of relativity,^[35] postulated the constancy and finiteness of the speed of light for all observers. He showed that this postulate, together with a reasonable definition for what it means for two events to be simultaneous, requires that distances appear compressed and time intervals appear lengthened for events associated with objects in motion relative to an inertial observer.

Einstein showed that if time and space is measured using electromagnetic phenomena (like light bouncing between mirrors) then due to the constancy of the speed of light, time and space become mathematically entangled together in a certain way (called Minkowski space) which in turn results in Lorentz transformation and in entanglement of all other important derivative physical quantities (like energy, momentum, mass, force, etc) in a certain 4-vectorial way (see special relativity for more details).

Classical mechanics

Newton's Second Law

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Classical mechanics

In classical mechanics Newton's concept of "relative, apparent, and common time" can be used in the formulation of a prescription for the synchronization of clocks. Events seen by two different observers in motion relative to each other produce a mathematical concept of time that works pretty well for describing the everyday phenomena of most people's experience.

Modern physics

In the late nineteenth century, physicists encountered problems with the classical understanding of time, in connection with the behavior of electricity and magnetism. Einstein resolved these problems by invoking a method of synchronizing clocks using the constant, finite speed of light as the maximum signal velocity. This led directly to the result that time appears to elapse at different rates relative to different observers in motion relative to one another.

Two-dimensional space depicted in three-dimensional spacetime. The past and future light cones are absolute, the "present" is a relative concept different for observers in relative motion.

Spacetime

Main article: Spacetime

Modern physics views the curvature of spacetime around an object as much a feature of that object as are its mass and volume.^{[citation needed]}

Time has historically been closely related with space, the two together comprising spacetime in Einstein's special relativity and general relativity. According to these theories, the concept of time depends on the spatial reference frame of the observer, and the human perception as well as the measurement by instruments such as clocks are different for observers in relative motion. Even the temporal order of events can change, but the past and future are defined by the backward and forward light cones, which never change.^{[citation needed]} The past is the set of events that can send light signals to the observer, the future the events to which the observer can send light signals. All else is non-observable and within that set of events the very time-order differs for different observers.^{[citation needed]}

Time dilation

Relativity of simultaneity: Event B is simultaneous with A in the green reference frame, but it occurred before in the blue frame, and will occur later in the red frame.

Main article: Time dilation

"Time is nature's way of keeping everything from happening at once". This quote, attributed variously to Einstein, John Archibald Wheeler, and Woody Allen, says that time is what separates cause and effect. Einstein showed that people travelling at different speeds, whilst agreeing on cause and effect, will measure different time separations between events and can even observe different chronological orderings between non-causally related events. Though these effects are minute unless one is travelling at a speed close to that of light, the effect becomes pronounced for objects moving at speeds approaching the speed of light. Many subatomic particles exist for only a fixed fraction of a second in a lab relatively at rest, but some that travel close to the speed of light can be measured to travel further and survive much longer than expected (a muon is one example). According to the special theory of relativity, in the high-speed particle's frame of reference, it exists, on the average, for a standard amount of time known as its mean lifetime, and the distance it travels in that time is zero, because its velocity is zero. Relative to a frame of reference at rest, time seems to "slow down" for the particle. Relative to the high-speed particle, distances seems to shorten. Even in Newtonian terms time may be considered the fourth dimension of motion; but Einstein showed how both temporal and spatial dimensions can be altered (or "warped") by high-speed motion.

Einstein (The Meaning of Relativity): "Two events taking place at the points A and B of a system K are simultaneous if they appear at the same instant when observed from the middle point, M, of the interval AB. Time is then defined as the ensemble of the indications of similar clocks, at rest relatively to K, which register the same simultaneously."

Einstein wrote in his book, Relativity, that simultaneity is also relative, i.e., two events that appear simultaneous to an observer in a particular inertial reference frame need not be judged as simultaneous by a second observer in a different inertial frame of reference.

time space

Spacetimes are the arenas in which all physical events take place—an event is a point in spacetime specified by its time and place. For example, the motion of planets around the sun may be described in a particular type of spacetime, or the motion of light around a rotating star may be described in another type of spacetime. The basic elements of spacetime are events. In any given spacetime, an event is a unique position at a unique time. Examples of events include the explosion of a star or the single beat of a drum.

A spacetime is independent of any observer.[3] However, in describing physical phenomena (which occur at certain moments of time in a given region of space), each observer chooses a convenient coordinate system. Events are specified by four real numbers in any coordinate system. The worldline of a particle or light beam is the path that this particle or beam takes in the spacetime and represents the history of the particle or beam. The worldline of the orbit of the earth is depicted in two spatial dimensions x and y (the plane of the earth's orbit) and a time dimension orthogonal to x and y. The orbit of the earth is an ellipse in space alone, but its worldline is a helix in spacetime.

The unification of space and time is exemplified by the common practice of expressing distance in units of time, by dividing the distance measurement by the speed of light.

[edit] Space-time intervals

Spacetime entails a new concept of distance. Whereas distances in Euclidean spaces are entirely spatial and always positive, in special relativity, the concept of distance is quantified in terms of the space-time interval between two events, which occur in two locations at two times:
s^2 = c^2\Delta t^2 - \Delta r^2\, (spacetime interval),

where:

c is the speed of light,
Δt and Δr denote differences of the time and space coordinates, respectively, between the events.

(Note that the choice of signs for s2 above follows the Landau-Lifshitz spacelike convention. Other treatments reverse the sign of s2.)

Space-time intervals may be classified into three distinct types based on whether the temporal separation (c2Δt2) or the spatial separation (Δr2) of the two events is greater.

Certain types of worldlines (called geodesics of the spacetime) are the shortest paths between any two events, with distance being defined in terms of spacetime intervals. The concept of geodesics becomes critical in general relativity, since geodesic motion may be thought of as "pure motion" (inertial motion) in spacetime, that is, free from any external influences.

[edit] Time-like interval

\begin{align} \\ c^2\Delta t^2 &> \Delta r^2 \\ s^2 &> 0 \\ \end{align}

For two events separated by a time-like interval, enough time passes between them for there to be a cause-effect relationship between the two events. For a particle traveling at less than the speed of light, any two events which occur to or by the particle must be separated by a time-like interval. Event pairs with time-like separation define a positive squared spacetime interval (s2 > 0) and may be said to occur in each other's future or past.

The measure of a time-like spacetime interval is described by the proper time:
\Delta\tau = \sqrt{\Delta t^2 - \frac{\Delta r^2}{c^2}} (proper time).

The proper time interval would be measured by an observer with a clock traveling between the two events in an inertial reference frame, when the observer's path intersects each event as that event occurs. (The proper time defines a real number, since the interior of the square root is positive.)
[edit] Light-like interval

\begin{align} c^2\Delta t^2 &= \Delta r^2 \\ s^2 &= 0 \\ \end{align}

In a light-like interval, the spatial distance between two events is exactly balanced by the time between the two events. The events define a squared spacetime interval of zero (s2 = 0).

Events which occur to or by a photon along its path (i.e., while travelling at c, the speed of light) all have light-like separation. Given one event, all those events which follow at light-like intervals define the propagation of a light cone, and all the events which preceded from a light-like interval define a second light cone.

[edit] Space-like interval

\begin{align} \\ c^2\Delta t^2 &< \Delta r^2 \\ s^2 &< 0 \\ \end{align}

When a space-like interval separates two events, not enough time passes between their occurrences for there to exist a causal relationship crossing the spatial distance between the two events at the speed of light or slower. Generally, the events are considered not to occur in each other's future or past. There exists a reference frame such that the two events are observed to occur at the same time.

For these space-like event pairs with a negative squared spacetime interval (s2 < 0), the measurement of space-like separation is the proper distance:
\Delta\sigma = \sqrt{\Delta r^2 - c^2\Delta t^2} (proper distance).

Like the proper time of time-like intervals, the proper distance (Δσ) of space-like spacetime intervals is a real number value.