The concept of space and time as a four dimensional continuum, spacetime, is fundamental to Einstein's relativity. However, the mechanics of spacetime seem to be little understood outside the scientific community.
It's not simply a matter of incorporating time as a dimension equivalent to the three spatial ones. It is more complex than that.
So this section explains:
The mechanics of spacetime is fundamentally based on the concept of the spacetime interval. This may be described loosely as the "distance" in four dimensional spacetime between one event and another.
Spacetime diagrams are used to express spacetime relationships. The simplest way of showing this is with time shown as the vetical axis and spatial distances horizontally. The time interval and spatial distance can then be combined if we use appropriate units. One way to do this is by measuring time in seconds, and distances in light-seconds (i.e. the distance that light travels in a second).
An event A (person A clapping) is separated in space and time from person B clapping, and from person C clapping. In the case of B, the time interval is larger than the spatial distance, but for C the spatial distance is larger than the time interval, as shown on the diagram. However, this does not properly represent the relationship because spatial distances and time intervals do not combine in the manner that you might expect:
Spatial distances combine according to the Pythagorean theorem. I.e.:
d2 = x2 + y2 + z2
where:
d is the spatial distance (in light-seconds).
x, y and z are the distances in the three spatial dimensions (in light-seconds).
But, spatial distances are subtracted from the time intervals to derive the spacetime interval. I.e.
s2 = t2 - d2
where:
s is the spacetime interval (in seconds or light-seconds).
t is the time interval (in seconds).
d is the spatial distance (in light-seconds).
So the relationship between event A and B, for which the time interval is greater than the spatial interval, is said to be "timelike", because, when the spatial interval is subtracted from the time interval, a time interval (AB') remains. Whereas the relationship between event A and C, for which the spatial interval is greater than the time interval, is said to be "spacelike", as a spatial interval remains when they are combined (the distance AC'). Between the timelike and spacelike areas lies a line at 45o for which the time interval and spatial interval is the same at each point. This marks the boundary between the timelike and spacelike intervals.
In Spacetime Mechanics we've shown the spacetime diagram two dimensionally, but it is often represented three dimensionally, with time shown as the vertical (z) axis and space represented by the horizontal (x and y) axes. I.e. One of the spatial dimensions is suppressed to allow time to be included in the 3D representation.
In this 3D diagram, the line dividing the timelike and spacelike intervals becomes a double cone. The front cone stands point downwards on top of the 2D plane representing 3D space. The back cone stands with its point upwards, below the plane. Event A is at the point where the two cones meet the plane.
Events within the front cone are timelike (with respect to A) and could be caused by event A. Events in the back cone are also timelike (with respect to A) and are past events that could have caused event A. Events outside the cones are spacelike (with respect to A) and can neither cause, nor be caused by, event A.
The surface of the two cones represent the spacetime points where the spatial and time intervals are exactly the same (with respect to event A). We noted above, that the spatial distance is subtracted from the time interval to arrive at the spacetime interval. Hence, this has the strange effect that, the spacetime interval at each point on the surface of the cones is zero.
As noted above, the surface of the two cones in the 3D diagram represent the spacetime points where the spatial and time intervals are exactly the same. This has the strange effect that, the spacetime interval at each point on the surface of the cones is zero (with respect to event A).
It is very easy to misinterpret the meaning of a zero spacetime interval. What it signifies is that light travelling towards event A would move radially along the surface of the back cone, until it reached event A. Similarly, light travelling away from event A would move radially along the surface of the front cone until it reached its target (event B). In both cases, the resulting spacetime interval is zero. So, to light, there is no time passage, nor spatial distance, between it's emission and reception. I.e. They happen at the same time and the same place, in the rest frame of the light. For this reason, the cones are often referred to as light cones.
We have shown the light cones at 45 degrees to the 2D surface representing the present, and they are quite distinct from the present, and from each other. However, for light, the cones are flattened onto each other by the lack of any time lapse. This also superimposes them on the present. That may have led to many misunderstandings about the present in spacetime, as it is only true for light (and other electromagnetic phenomena). Macro objects experience the universe very differently from this, as is displayed in the following example...
Two observers, Alfie and Betty, have a flashlight each and are mutually at rest one light-second apart. Alfie flashes his torch, and this is event A. Betty flashes her torch a second later, and this is event B. To light passing from event A to event B, these events are simultaneous and co-located.
But Alfie and Betty do not experience them in that manner. A second after Betty flashes her torch Alfie will see the flash. As Betty is a light-second distant from him, Alfie knows that the light took a second to reach him. Hence Alfie concludes that event B happened a second after event A. So Alfie experiences the flashes of light as happening a second apart in time, and a light-second apart in distance.
This means that zero spacetime intervals do not necessarily mean that:
To macro objects, a zero spacetime interval just means that the spatial and temporal "distances" beween two events are the same magnitude. So two events could be co-located and simultaneous, or be a thousand light-years distant and separated by a thousand years in time.
Furthermore, if this is true for zero intervals, it is also true for non-zero intervals. E.g. If a second after Betty flashes her torch, she flashes it again (event C):
Hence, it would appear that the property of spacetime whereby time intervals and spatial distances cancel does not describe how macro objects experience space and time.
This problem with the interpretation of spacetime intervals may lie at the root of popular myths about spacetime, such as the Andromeda Paradox. In this the fact that events far distant in time may have a zero spacetime interval with respect to an event on the Earth, if they are similarly distant in space (such as from the Earth to Andromeda), is taken to mean that they are simultaneous. As we have noted, this is only true for light and other electromagnetic phenomena. It does not accord with the way that macro objects experience spacetime.
Coordinate time is the time interval that a perfect clock at rest in a given frame of reference would measure between any two events. It is called "coordinate" time because the time interval is frame dependent, i.e. it can be different when measured in different frames of reference. This effect is described in the Twin Paradox thought experiment, where one twin (who travels at near-light velocity away from and then back to the Earth) ages slower than his twin (who remains on the Earth). This may sound incredible, but the effect is fundamental to Einstein's theory, and has been empirically proven to exist.
However, because the time elapse can be different for different observers, it is sometimes mistakenly thought that it is subjective, i.e. an effect of the perception of an individual. This is incorrect. The coordinate time between two events is the same for any clocks and observers mutually at rest in a given frame, even if they are, in principle, millions of miles apart spatially. So coordinate time is objective, even though it is frame dependent. It is not a subjective effect dependent upon an individual observer. According to Einstein's theory, there really is no absolute passage of time.
Proper time is a name given to a "timelike" spacetime interval between two events that would be recorded by a clock that is co-located with the events. Under special relativity, there is only one frame of reference in which the events are co-located. So the proper time interval between them is unique, and because the events are co-located, it is frame invariant. It is also equal to the coordinate time that a clock at rest in that frame of reference would record.
Under general relativity the proper time depends upon the path taken by the clock between the two events. There are many paths that can be taken, therefore the proper time interval is not unique. However, for any given path it is frame invariant, and is the same as the coordinate time that an observer would experience in taking that path between the events.
So is proper time a complete and absolute measure of time? Well, we noted above under coordinate time that there is no absolute passage of time. So proper time is a consistent measure of time. However, it achieve that consistency by being a sub-set of the coordinate time relationships between events. E.g. The proper time interval of 7 seconds measured by a clock at rest in the frame of reference in which two events are co-located equates to:
So the proper time equates to the coordinate time measured in the same rest frame, or for the same path between the events. However, there are myriad other coordinate distance and time relationships between the same events that equate to that proper time interval.
The model of space and time as a four dimensional continuum, spacetime, is fundamental to Einstein's relativity, and is very successful in both according with and predicting empirical observations. Furthermore, the fact that spacetime intervals are frame invariant presents a more coherent picture of space and time than the frame dependent coordinate distances and time intervals. Indeed, it is so successful that there seems to be a growing belief that spacetime is both real and a complete description of the spatio-temporal relationships between events. However, spacetime is not without issues:
Hence, since coordinate relationships are real, spacetime intervals are also real, but are a sub-set of the coordinate space-time relationships.
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