Space-time mathematical models

In both the special and general theories of relativity, neither space by itself nor time by itself is independent of the state of the observer. Only a certain mathematical union of them, called space-time, has invariant properties. The geometry of space-time is the basis for relativistic physics, which is seen in our solar system. A full description of the advancement of the perihelion of Mercury, for example, requires the use of relativity. Also, the way the mass of the Sun can bend light coming from other stars and galaxies is described by relativity theory.

Overview

Space-time is a four-dimensional coordinate system or reference frame in which one mathematically describes the spatial location and temporal coordinate of an event. Such a frame of reference can be either inertial or non-inertial. An inertial frame of reference is often defined in Newtonian mechanics as a frame that is not accelerated, but then one must ask, “Accelerated relative to what?” A better definition consistent with relativity is that a local inertial frame (or LIF) is a frame in which a body subject to no external force moves at constant speed in a straight line, or alternatively, it is a frame in which everything is weightless. The word local is added because, in the space-time of general relativity, it is not possible to have a truly universal inertial frame if any mass is present.

A wide variety of experiments have repeatedly confirmed that physical phenomena do not fundamentally differ from one inertial frame of reference to any other. The special theory of relativity asserts this as a basic postulate or principle: all the laws of physics are the same in every inertial frame of reference. This means that both the mathematical form of fundamental equations of physics and the values of the physical constants that they contain are the same in all inertial frames. When this principle is applied to the theory of electromagnetism, it requires that observers in all inertial frames of reference agree about the numerical value of the speed of electromagnetic waves in empty space. The universality of this speed—henceforth referred to as the speed of light and represented by the letter c—requires that space and time separately cannot be invariant but must change upon transformation from one inertial frame of reference to another.

An event is the name given to something that happens at a particular time and place. The collection of all events (the whens and wheres) in the history of a particle is called its world line. Measurements of the time and place of an event will vary from one inertial frame to another, but there are equations called Lorentz transformations that relate the time and space coordinates of an event in one inertial frame to the time and space coordinates in another inertial frame, based on the relative velocity of the two frames. All the famous phenomena predicted by the special theory of relativity (relativity of simultaneity, length contraction, time dilatation) can be derived from these transformations.

Although the space and time coordinates of an event will differ from one inertial frame to another, there is a quantity called the space-time separation (or space-time interval) between the events that is invariant, meaning that it has the same value in all inertial frames. Let A and B be two events with time and space coordinates (t_A, x_A, y_A, z_A) and (t_B, x_B, y_B, z_B) in one inertial frame, and (t_A′, x_A′, y_A′, z_A>′) and (t_B′, x_B′, y_B′, z_B′) in another inertial frame. The space-time separation s between the two events is defined as

s² = -c² (t_B - t_A)² + (x_B - x_A)² + (y_B - y_A)² + (z_B - z_A)²

as calculated in one frame and

s² = -c² (t_B′ -t_A′)² + (x_B′ - x_A′)² + (y_B′ - y_A′)² + (z_B′ - z_A′)²

as calculated in the other frame. It can be shown using the Lorentz transformations that these two expressions for s are equivalent, demonstrating that the space-time separation is invariant between inertial frames.

The interval or separation between two events in space-time is somewhat analogous to the distance r between two points in space:

r² = (x_B - x_A)² + (y_B - y_A)² + (z_B - z_A)²

However, there is a fundamental difference between the geometry of space-time and the geometry of space by itself. Notice that the square of the difference in the time coordinates appears in the space-time formula with the opposite sign from the squares of the differences in the spatial coordinates.

Applications

Because the squares of the time and space coordinate differences have opposite signs in the formula for space-time interval or separation, the square of the interval or separation between two distinct space-time events can be positive, zero, or negative, depending on how the squared difference of the time coordinates compares to the sum of the squared differences of the space coordinates.

If the squared interval or separation is positive, the squared difference of the space coordinates dominates over the squared difference of the time coordinates, and the separation is called space-like. This means that there exists some inertial frame of reference in which the time coordinates of the two events are equal, so in this frame, the two events are simultaneous, and they differ only in spatial location. In particular, neither event can be the cause or effect of the other since all physical influences require time to propagate. In other inertial frames, the events will have different time coordinates, and it is possible for either of the two events to have a larger time coordinate, meaning it occurred later. Consequently, “later” and “earlier” have no universal meaning for a pair of events with a space-like separation since, in some frames, one event occurred later while in other frames the other event occurred later.

If the squared interval or separation is negative, the squared difference of the time coordinates dominates over the squared difference of the space coordinates, and the separation is called time-like. This means that there is some inertial frame in which the spatial coordinates of the two events are equal, so in this frame, the two events occurred at the same location and differ only in time. When a pair of events have a time-like separation, observers in all inertial frames agree on which event occurred first and which occurred second.

If the squared interval or separation is zero, the squared difference of the space coordinates equals the squared difference of the time coordinates (multiplied by the speed of light squared), and the separation is called null or light-like. This means that, in every inertial frame, the pair of events may be connected by the world line of a ray of light moving from one to the other. Such a ray of light could be the agent by which the earlier event causes the later event. Watching a pair of events that have light-like separation, observers in all inertial frames agree on which event occurred first and which occurred second.

Since a particle is always at its own location, intervals between events on the world line of a particle with mass must be time-like. On the other hand, intervals between events on the world line of a photon (a “particle” of light) must be light-like or null.

The sign of the squared intervals between events can be used to divide space-time into regions of different character. Suppose that event A is at the coordinate origin (“here and now”) of space-time (x_A = 0, y_A = 0, z_A = 0, t_A = 0). As an aid to visualization, the z coordinate may be suppressed; then, it is possible to draw a diagram illustrating these regions. The surface mapped out by all null intervals is a double cone. The upper cone (positive t) represents all events in the future that can be reached by a light ray emitted here and now, while the lower cone (negative t) represents all events in the past that could have sent a light ray to arrive here and now. All events that occur within the cone have squared space-time separations from event A that are negative or time-like.

Consequently, the world line of any particle with mass that passes through (coincides with) the event chosen as the here and now”origin of the figure is confined to the interior of the light cone. All events outside the cone have squared space-time separations from event A that are positive or space-like. Consequently, they can neither influence nor be influenced by event A, for to do so, the influence would have to travel between event A and any event outside the cone at a speed greater than the speed of light.

The time axis of any space-time coordinate plot indicates the passage of time in the frame of reference with those coordinates. In an inertial frame, the time axis will be a straight line. The world line, curved or straight, of any particle, can be considered a time axis for that particle. Intervals along its world line define its proper time, which elapses on a clock carried by the particle.

The straight line between two points in Euclidean space has the shortest length of any curve joining them. The geometry of space-time is such that the straight line between two events with a time-like separation has the longest proper time of any world line joining them. This is the basis of a straightforward prediction of relativity, which is usually called the twin paradox. Effects of the special theory of relativity are analyzed from inertial frames of reference but may include accelerated objects, such as the twin who travels out and back, thus aging less than the twin remaining at rest in one inertial frame throughout the other’s trip.

Analysis of motion from the point of view of observers in accelerated frames of reference is also possible, but it uses the mathematical concepts of differential geometry. As seen from accelerated frames of reference, the structure of space-time is not globally covariant but only locally covariant. This means that the light cones at various events may be tilted in relation to each other. Albert Einstein’s general theory of relativity attributes such distortion of the geometry of space-time to gravity. This theory can be summarized in two intimately linked statements: (1) matter warps space-time, and (2) warped space-time tells matter and light how to move.

Context

The root of the concept of space-time was the discovery by Hendrik Antoon Lorentz, published in 1898, of the rules of transformation of the coordinates of an event from one inertial frame of reference to any other inertial frame of reference. His derivation was carried out to find a transformation that does not change the form of the fundamental laws of electrodynamics, known as Maxwell’s equations. However, Lorentz did not claim that the transformations he found that kept electromagnetism invariant had the broad applicability they are now understood to have. It remained for Einstein to formulate a comprehensive view, published in 1905 in his special theory of relativity, of space and time as measured in inertial frames moving relative to each other and their dependence on the state of motion of an observer.

Even as he was establishing the foundations of what is now called the special theory of relativity, Einstein was aware of the incompatibility of these ideas with Newtonian gravitational theory. His early work on extending the principle of relativity beyond inertial frames of reference was hampered by mathematical complexities. Hermann Minkowski, a former math professor of Einstein at Zurich Polytechnic University, in an address presented in 1908, introduced the concept of unified space-time based on the new ideas expressed in Einstein’s 1905 description of his special theory of relativity. Minkowski realized that Einstein’s assertion of the constancy of the speed of light for all observers implied that space and time were fundamentally linked. He also introduced the powerful techniques of geometry, which provided both the mathematical formalism for dealing with non-inertial frames as well as intuitive insights into the physical implications. By exploiting ideas first introduced to understand the differential geometry of curved surfaces, unified space-time became the natural way to understand all of physics. Thus, the mathematics of non-Euclidean geometry found application in Einstein’s general theory of relativity, which was published in 1916. The general theory is a comprehensive synthesis of the relations among space, time, matter, and motion from the point of view of any frame of reference whatsoever.

In contemporary physics, space-time is accepted as the arena in which all things exist and move. The assertion that the laws of physics must be independent of the arbitrary choice of a particular frame of reference in space-time is a powerful working tool of the theoretical physicist. This requirement puts limits on possible new hypotheses and the equations they imply in almost all areas of physics. The one branch of physics that has remained at odds with relativity theory is quantum mechanics. It remains to be seen whether quantum behavior can be unified with the space-time of relativity.

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