Space-time

Type of physical science: Relativity

Field of study: Special relativity

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.

Overview

An inertial frame of reference (one in which the motion of a body subject to no external influence is at constant speed in a straight line) simplifies the analysis of motion. Any frame of reference that has constant velocity (unchanging speed and direction of motion) with respect to an inertial frame of reference is also an inertial frame. An enormous quantity of experimental evidence has confirmed that physical phenomena do not fundamentally distinguish one inertial frame of reference from any other. The principle of relativity asserts that 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 numerical 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" (for "constant")--requires that space and time separately cannot be invariant, but must change upon transformation from one inertial frame of reference to another. A unique rule of coordinated changes (covariance) relates the observations of location in space and time of occurrence of an event made from various frames of reference, to ensure that the speed of light is always observed to have the same value. This Lorentz transformation law, relating the space and time coordinates measured from various frames of reference, implies that the "where" and "when" of events are aspects ("components") of a unified structure, which is called "space-time."

One way to approach an understanding of space-time is through study of how its components, as measured in various frames of reference, are related. This is analogous to the study of vectors through the rules that relate their components in various coordinate systems. For example, consider displacements on the surface of the earth. These displacements may be described by northward and eastward coordinates based on the local sighting of Polaris, the North Star, and sense of rotation of the earth. Alternately, north could be specified by the local pointing of a magnetic compass. At most sites, there is some disagreement between these two systems, so that a displacement that is purely northward magnetically may have an east (or west) component astronomically, and its astronomical north component will, in general, be of a different magnitude from that of its magnetic north value. The "discrepancy" between the two systems, resulting from rotation of axes of one in relation to the other, is so familiar as to no longer be surprising. The different values of vector components do not provide a reason to reject either system. Rather, the user of geometry needs to become familiar with the rule that relates measurements made in the two systems, so that an observer using either rule can compare information with an observer using the other. Similarly, suppose that space coordinates, xa and xb, and time coordinates, ta and tb, are assigned to a pair of events, a and b, by an observer in one inertial frame of reference. An observer in another inertial frame will, in general, measure different space coordinates, xa¹ and xb¹, and different time coordinates, ta¹ and tb¹, for the same pair of events. The Lorentz transformation law connects the descriptions, so that knowledge of either set of space-time coordinates and the relative velocity of the frames of reference allows prediction of the other set of space-time coordinates. The famous phenomena predicted by the special theory of relativity (relativity of simultaneity, length contraction, time dilatation) can all be derived from this transformation.

Another way to approach an understanding of a geometrical structure, whether familiar Euclidean space or space-time, is through the study of the quantities, calculated from components, that are the same in all coordinate systems. For example, using Cartesian coordinates x and y in a plane produces components of displacement between two points, a and b, whose values depend on the orientation of the coordinate axes. Yet, there is a mathematical expression that can be formed, the square of the distance between two points, which has the same mathematical form and the same numerical value for a given pair of points in all coordinate systems that agree on units of length for all axes. This quantity, (xa - xb)² + (ya - yb)² = (xa' - xb')² + (ya' - yb')² is called an invariant. The mathematical forms of the invariants of a geometrical structure characterize that structure. The form of the distance squared indicates that circles are invariant under rotation of Cartesian axes in a plane and provide a way to compare coordinate axes in different systems to ensure that they all use the same unit of measure for length. To construct an invariant in space-time, one must measure the differences in time coordinates in the same units as differences in space coordinates. This is possible precisely because of the existence of a universal conversion factor (which multiplies units of time to produce units of length) on the speed of light. The space-time analogue of the squared distance between points in space is the quantity (xa - xb)² - o²(ta - tb)² = (xa' - xb')² - c² (ta' - tb')² called the square of the interval between the events a and b.

There is a similarity with space geometry, since this formula also involves squares of coordinate differences. There is a fundamental distinction between the geometry of space-time and that of space, however, since squares of time coordinate differences enter the formula for the squared interval with the opposite sign from the squares of space coordinate differences. The form of the interval squared indicates that hyperbolas are invariant under "rotation" of axes in space-time.

The magnitude of this "rotation" corresponds to the relative speed of the two inertial frames of reference being compared.

Applications

There is a profound distinction between space-time geometry implied by the squared interval and space geometry implied by the squared distance. Squared distance is positive if the two space points are distinct, while the squared interval between two distinct events can be positive, zero, or negative.

If the squared difference of space coordinates dominates over the squared difference of time coordinates (converted to units of length), one can say that the interval between the pair of events under discussion is spacelike. This means that there is some inertial frame of reference in which the time coordinates of the events are equal (for this observer, the events happen "at the same time"), and may be described as being simply "elsewhere" in relation to the other. In particular, neither can be the cause or effect of the other, since all physical influences require time to propagate. Indeed, in some other inertial frames of reference, the events will be assigned different time coordinates, and it is possible for either of the two events to have the larger coordinate. Thus, the ideas of "later" and "earlier" have no intrinsic meaning for a pair of events whose separation is spacelike.

If the squared difference of space coordinates equals the squared difference of time coordinates, the interval is null (zero) and is called lightlike. This means that in every inertial frame of reference the pair of events under discussion may be connected by passage of a ray of light from one to the other. Such a ray of light could be the agent by which one event is the cause of the other. Most physicists assume that causes must precede their effects in time, as measured in every observer's frame of reference. The agreement of all inertial observers on the sign of the difference in time coordinates for any pair of events whose interval is null requires that no observer's speed can achieve or exceed the speed of light as measured from any other observer's frame of reference.

If the squared difference of time coordinates dominates over the squared difference of space coordinates, one can say that the interval between the pair of events under discussion is timelike. This means that there is some inertial frame of reference in which the space coordinates of the events are equal (the events happen "in the same place"), and may be described as one being in the future of the other. In particular, they may be two events in the history of a particle with mass, which all inertial observers agree moves at speeds less than the speed of light.

The invariant signs of the squared intervals between events can be used to divide space-time into regions of different character. For simplicity, suppose that event a is at the origin ("here and now") of space-time. Space is three-dimensional and space axes can be rotated among various inertial observers. Thus, the interval from the origin to any other event x² + y² + z² - c² t² = x² + y² + z² - c² t² may involve different y and z coordinates in different frames. If, as an aid to visualization, the z coordinates are suppressed (set z = z' = 0), it is possible to draw a diagram illustrating these regions. The surface mapped out by all null intervals is a cone. The upper branch (positive t) represents all events in the future that can be reached by a single light ray emitted here and now, while the lower branch (negative t) represents all events in the past that could have sent a single light ray to arrive here and now. The collection of all events in the history of a particle is called its world line. Since a particle is always at its own location, intervals between events on a world line of a particle with mass must be timelike. Thus, the world line of any such particle that passes through ("coincides with") the event chosen as the origin of the figure is confined to the interior of this light cone. For any such particle now, the region outside the cone is "elsewhere." Because of the invariance of interval, this division of the regions of space-time in relation to the event chosen as the origin is absolute.

Although the time axis of any inertial frame of reference will be a straight line inside the light cone of the figure, the world line, curved or straight, of any particle can be considered a time axis for that particle. Intervals along the world line define "proper time," which elapses on a clock carried by the particle. Just as the straight line between two points in Euclidean space has the shortest length of any curve joining them, so the straight line between two timelike separated events in space-time 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 observation of accelerated objects, such as the twin which travels out and back, thus aging less than the one remaining at rest in one frame throughout their separation. Analysis of motion from the point of view of observers in accelerated frames of reference is also possible, but 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: Matter warps space-time, and warped space-time moves matter.

Context

The root of the concept of space-time is 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 that transformation which does not change the form of the fundamental laws of electrodynamics, known as Maxwell's equations. Yet, Lorentz himself did not claim for himself the transformation that he found the broad applicability as it is now understood it has. It remained for Einstein to formulate a comprehensive view, published in 1905, of the relations between space and time coordinates, and their dependence on the state of motion of an observer.

Even as the foundations of what is now called the special theory of relativity were being articulated, Einstein was aware of the incompatibility of Newtonian gravitational theory with these ideas. His early work on extending the principle of relativity beyond the limitation to inertial frames of reference was hampered by mathematical complexities. Hermann Minkowski's introduction, described in an address presented in 1908 of the powerful techniques of geometry, provided a most valuable means of building insight. His space-time, though fundamentally different from the familiar space of Euclid, has many analogies with it. The readjustment to accommodate the possibilities of negative or zero squared intervals in addition to positive ones rewards the student with an intuitive understanding of how relations between lengths and times in the special theory of relativity depend on the inertial frame of reference chosen. Furthermore, by exploiting ideas first introduced to understand the differential geometry of curved surfaces, the space-time view becomes the natural way to understand all of physics. Thus, the mathematics of non-Euclidean geometry finds application in Einstein's creation of the general theory of relativity, which was published in 1916. This is a comprehensive classical 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 requirement that the laws of physics be independent of arbitrary choice of a particular frame of reference in space-time is a powerful working tool of the theorist. This assumption limits the forms of candidates for new fundamental equations in all areas of research, even in the description of quantum behavior in the subatomic realm at the highest observed energies.

Principal terms

COVARIANT: interdependent according to a particular mathematical rule, such as that which connects the space coordinates and time of an event in relativity

EVENT: a fundamental "point" of space-time, specified not only by a place but also by a time of occurrence

INTERVAL: a measure of the separation in space-time between two events; intervals may be timelike, spacelike, or null (lightlike)

INVARIANT: unchanged by a transformation of coordinates, such as the interval between two events in space-time

WORLD LINE: a curve in space-time representing all events in the history of a particle; world lines of particles with mass are timelike

Bibliography

Ferington, Esther. THE COSMOS. Alexandria, Va.: Time-Life Books, 1988. This profusely illustrated large-format book is a brief but surprisingly comprehensive introduction to modern cosmology. Includes illuminating presentations on the concept of space-time in both the special and the general theories of relativity.

Minkowski, Hermann. "Space and Time." In THE PRINCIPLE OF RELATIVITY. New York: Dover, 1952. This early presentation of the space-time concept to a technical audience uses for the most part elementary mathematics to explore some implications of the relativistic unity of space and time.

Misner, Charles W., Kip S. Thorne, and John A. Wheeler. GRAVITATION. San Francisco: W. H. Freeman, 1973. This comprehensive textbook on general relativity begins with a clear exposition of the space-time concept of special relativity in a form that facilitates its natural extension to general relativity. Suitable for the diligent layperson with some background in physics, though many later parts are written at more advanced mathematical levels.

Schwinger, Julian. EINSTEIN'S LEGACY: THE UNITY OF SPACE AND TIME. New York: Scientific American Books, 1986. A lively and well-illustrated account of the special and general theories of relativity, this volume includes a careful elementary presentation on the concept of space-time with helpful examples and applications.

Taylor, Edwin F., and John A. Wheeler. SPACE-TIME PHYSICS. San Francisco: W. H. Freeman, 1966. This brief text is the most thorough treatment of the subject at an elementary mathematical level. Careful reading and working through its examples develops intuition for thinking relativistically. The few problems that require calculus for their solution are clearly identified.