# TIME SUPPLEMENT _03

1. How Does Gravity Affect Time?

Einstein’s general theory of relativity (1915) is a generalization of his special theory of special relativity (1905). It is not restricted to inertial frames, and it encompasses a broader range of phenomena, namely gravity and accelerated motions. According to general relativity, gravitational differences affect time by dilating it. Observers in a less intense gravitational potential find that clocks in a more intense gravitational potential run slow relative to their own clocks. People live longer in basements than in attics, all other things being equal. Basement flashlights will be shifted toward the red end of the visible spectrum compared to the flashlights in attics. This effect is known as the gravitational red shift. Even the speed of light is slower in the presence of higher gravity.

Informally one speaks of gravity bending light rays around massive objects, but more accurately it is the space that bends, and as a consequence the light is bent, too.  The light simply follows the shortest path through spacetime, and when space curves the shortest paths are no longer Euclidean straight lines.

1. What Happens to Time Near a Black Hole?

There is a massive black hole at the center of our galaxy in the direction of the constellation Sagittarius. A black hole is a volume of very high gravitational field or severe warp in the spacetime continuum. Astrophysicists believe black holes are commonly formed by the inward collapse of stars that have burned out, stars that are at least two and a half times larger than our sun. The center of a black hole (sometimes called “the singularity”) is extremely dense—infinitely dense according to relativity theory, but not that dense according to theories of quantum gravity. The hole is surrounded by an event horizon, marking the point of no return. Anything getting that close could never escape the inward pull, even if it had an unlimited fuel supply and could travel at near the speed of light. Anything falling into the hole by crossing the event horizon will quickly crash into the center of the black hole and be crushed to a point. The first black hole solution to Einstein’s equations of general relativity was discovered by Schwarzschild in 1916.  Because even light itself could not escape from inside a black hole, John Wheeler chose the name “black hole.”

In relativity theory, the proper time between two events along a worldline is the time that would be shown on a clock whose path in spacetime is that worldline between the events. The proper time is not the same as the coordinate time.  Coordinate time is time along the worldline of an ideal clock at the origin of the coordinate system. The coordinate time between the two events is the time separation of the events given by an observer at rest in the frame.  Proper time is independent of coordinate time, although the usual convention is to measure both times in the same units, namely seconds. As judged by a clock on earth in an earth-based frame of reference, an astronaut flying into a distant black hole will take an infinite coordinate time to reach the event horizon of the black hole. That is, if we could see the astronaut’s clock, the clock would appear to us to slow to a halt.  But as judged by the astronaut, it will take only a few microseconds of the astronaut’s proper time to pass through the event horizon and crash into the center of the black hole.

If you, the person falling toward the event horizon, were to successfully escape the pull towards the black hole and somehow return home, you’d discover that you were younger than your earth-bound twin and that your initially synchronized clocks showed that yours had fallen behind. It is in this sense that you would have experienced a time warp, a warp in the time component of spacetime.

1. What Is the Solution to the Twin Paradox?

This paradox, also called the clock paradox and the twin paradox, is an argument about time dilation that uses the theory of relativity to produce a contradiction. The argument considers two twins at rest with their clocks synchronized. One twin climbs into a spaceship and flies far away at a constant speed, then reverses course and flies back at the same speed. When they reunite, will the twins still be the same age? No. Relativity theory implies that the twin on the spaceship will return and be younger than the Earth-based twin.  The elapsed proper time of the twin who returns is less than the elapsed proper time of the Earth-based twin.  However, it’s all relative, isn’t it? That is, we could have considered the spaceship to be stationary. Wouldn’t relativity theory then imply that the Earth-based twin would race off (along with the Earth), then return and be the younger of the two? If so, we have a contradiction: when the twins reunite, each will be younger than the other.

Einstein worried about the paradox [Einstein, Naturwissenschaften, 6, 697 (1918)], and Herbert Dingle famously argued in the 1960’s that the paradox reveals an inconsistency in special relativity.  Almost all philosophers and scientists now agree that it is not a true paradox, in the sense of revealing a logical inconsistency within relativity theory, but is merely a complex puzzle that can be adequately solved within relativity theory.  The twin who feels the acceleration is the twin who becomes the younger twin, but the acceleration upon starting and reversing course is not what causes this difference in aging, and it is not essential to the paradox. The key idea is that there is an asymmetry in the two spacetime intervals taken by the twins between the goodbye event and the reunion. Sitting still on Earth is a way of maximizing the time between the events; flying fast in a spaceship is a way of minimizing the time between the events. The reasoning in the paradox makes the mistake of supposing that the twin on Earth could somehow have the shorter interval.

To explain that last point, let’s for simplicity’s sake assume the twin on Earth is fixed in an inertial frame. The way out of the paradox is to notice that the argument has two halves. The first half describes the twin in the spaceship flying away, then turning around and flying back to the Earth-based twin who remains fixed in an inertial frame during the flight. The second half describes the Earth-based twin as flying away from the spaceship and then returning to the spaceship, while the spaceship remains stationary in an inertial frame during the flight.  The problem is that in the second half of the paradox it was a mistake to suppose the spaceship “remains stationary in an inertial frame.” The spaceship’s frame can not be inertial. Also, because of the spaceship’s changing velocity by turning around, the twin on the spaceship has a shorter world-line than the Earth-based twin and takes less time than the Earth-based twin. The assumption is that the stars are not moving in tandem with the spaceship but are generally stationary relative to the Earth.  Without this crucial, but usually implicit, assumption, one couldn’t decide which twin was or was not in an inertial frame; if two twins are out in space alone with no stars (or other matter-energy), then when they meet again they will be the same age.

The production of the paradox depends on using a heuristic principle that the description of the world is equally valid from the point of view of any observer. That heuristic principle is embedded in the above remark, “…it’s all relative, isn’t it?”  The application of the principle makes the assumption that the two halves of the analysis are working with two equivalent descriptions of the same process. However, the two descriptions are not equivalent because, if there is an inertial frame in the first half of the argument, then there is no available inertial frame for the second half.

The analyst is always free to make the choice of a non-inertial frame in which the spaceship is considered to be stationary. This would complicate the analysis, and require general relativity instead of special relativity, but the result would be the same, namely no contradiction. So, it can’t be shown that the Earth-based twin is the younger, and the paradox is merely a puzzle that has a solution.

To dig more deeply into the cause of error in the reasoning, notice that the production of the paradox depends upon a careless use of the heuristic principle that the description of the world is equally valid from the point of view of any observer. This principle is misinterpreted in the twin paradox. What is always correct in relativity theory and what underlies the heuristic principle is the symmetry principle: the invariance of the laws under Lorentz transformations. These transformation equations give the relations between the coordinates of a single event [such as our spaceship’s flight away from the Earth-based twin] as measured by observers in two different inertial reference frames in motion relative to one another. But there aren’t two inertial frames to use in the case of the twin paradox, so the symmetry principle is correct, but the heuristic principle is not applicable. The argument of the twin paradox applies the heuristic principle anyway, and draws an incorrect conclusion that there is a contradiction.

What causes one twin to age differently? The best answer to this question is to re-examine the question itself. It was remarked above that the easiest way to see the dissimilarity in the two halves of the argument. Failure to notice the asymmetry in the two halves is the cause of the error in the paradox, but it’s not the cause of the age difference in the twins.  Their age difference isn’t caused by anything, just as light’s going at the speed of light instead of at some other speed isn’t caused by something but is just the way nature behaves, at least insofar as the theory of relativity is concerned.

1. What Is the Solution to Zeno’s Paradoxes?

See the article “Zeno’s Paradoxes” elsewhere in this encyclopedia.

1. How Do Time Coordinates Get Assigned to Points of Spacetime?

A space is a collection of points. In a space that is supposed to be time, these points are the instants. Our question is how we assign time numbers to these points. Before discussing time coordinates specifically, let’s consider what is meant by assigning coordinates to a mathematical space, one that might represent either physical space, or physical time, or spacetime, or something else. In a one-dimensional space, such as a curved line, we assign unique coordinate numbers to points along the line and we make sure that no point fails to have a coordinate. For a 2-dimensional space, we assign pairs of numbers to points. For a 3-d space, we assign triples of numbers. If we assign letters instead of numbers, we can’t use the tools of mathematics to describe the space. But we can’t assign any coordinate numbers we please. There are restrictions. For example, if the space has a certain geometry, then we have to assign numbers that reflect this geometry. Here is the fundamental method of analytic geometry:

Consider a space as a class of fundamental entities: points. The class of points has “structure” imposed upon it, constituting it a geometry–say the full structure of space as described by Euclidean geometry. [By assigning coordinates] we associate another class of entities with the class of points, for example a class of ordered n-tuples of real numbers [for a n-dimensional space], and by means of this “mapping” associate structural features of the space described by the geometry with structural features generated by the relations that may hold among the new class of entities–say functional relations among the reals. We can then study the geometry by studying, instead, the structure of the new associated system [of coordinates]. (Sklar, 1976, p. 28)

The goal in assigning coordinates to a space is to create a reference system for the space. A reference system is a reference frame plus either a coordinate system or an atlas of coordinate systems placed by the analyst upon the space to uniquely name the points. These names or coordinates are frame dependent in that a point can get new coordinates when the reference frame is changed. For 4-d spacetime obeying special relativity and its Lorentzian geometry, a coordinate system is a grid of smooth timelike and spacelike curves on the spacetime that assigns to each point three space coordinate numbers and one time coordinate number. No two distinct points can have the same set of four coordinate numbers. Inertial frames can have global coordinate systems, but in general we have to make due with atlases. If we are working with general relativity where spacetime can curve and we cannot assume inertial frames, then the best we can do is to assign a coordinate system to a small region of spacetime where the laws of special relativity hold to a good approximation. General relativity requires special relativity to hold locally, and thus for spacetime to be Euclidean locally. That means that locally the 4-d spacetime is correctly described by 4-d Euclidean solid geometry.  Consider two coordinate systems on adjacent regions. For the adjacent regions we make sure that the ‘edges’ of the two coordinate systems match up in the sense that each point near the intersection of the two coordinate systems gets a unique set of four coordinates and that nearby points get nearby coordinate numbers. The result is an “atlas” on spacetime.

For small regions of spacetime, we create a coordinate system by choosing a style of grid, say rectangular coordinates, fixing a point as being the origin, selecting one timelike and three spacelike lines to be the axes, and defining a unit of distance for each dimension. We cannot use letters for coordinates. The alphabet’s structure is too simple. Integers won’t do either; but real numbers are adequate to the task. The definition of “coordinate system” requires us to assign our real numbers in such a way that numerical betweenness among the coordinate numbers reflects the betweenness relation among points. For example, if we assign numbers 17, pi, and 101.3 to instants, then every interval of time that contains the pi instant and the 101.3 instant had better contain the 17 instant. When this feature holds, the coordinate assignment is said to be monotonic. There is no way to select one point of spacetime and call it the origin of the coordinate system except by reference to actual events. In practice, we make the origin be the location of a special event, and one popular choice is the birth of Jesus. Negative time coordinates are assigned to events occuring before the birth of Jesus.

The choice of the unit presupposes we have defined what “distance” means. The metric for a space specifies what is meant by distance in that space. The natural metric between any two points in a one-dimensional space, such as the time sub-space of our spacetime, is the numerical difference between the coordinates of the two points. Using this metric, the duration between the 11:00 instant and the 11:05 instant is five minutes. The metric for spacetime defines the spacetime interval between two spacetime locations, and it is more complicated than the metric for time alone. The spacetime interval between any two events is invariant or unchanged by a change to any other reference frame, although the spatial distances and durations do vary. More accurately, in the general theory, the infinitesimal spacetime interval between two neighboring points is invariant. The units of the spacetime interval are seconds squared.

In this discussion, there is no need to worry about the distinction between change in metric and change in coordinates. For a space that is topologically equivalent to the real line and for metrics that are consistent with that topology, each coordinate system determines a metric and each metric determines a coordinate system. More precisely, once you decide on a positive direction in the one-dimensional space and a zero-point for the coordinates, then the possible coordinate systems and the possible metrics are in one-to-one correspondence.

There are still other restrictions on the assignments of coordinate numbers. The restriction that we called the “conventionality of simultaneity” fixes what time-slices of spacetime can be counted as collections of simultaneous events. An even more complicated restriction is that coordinate assignments satisfy the demands of general relativity. The metric of spacetime in general relativity is not global but varies from place to place due to the presence of matter and gravitation. Spacetime cannot be given its coordinate numbers without our knowing the distribution of matter and energy.

The features that a space has without its points being assigned any coordinates whatsoever are its topological features. These are its dimensionality, whether it goes on forever or has a boundary, how many points there are, and so forth.

1. How Do Dates Get Assigned to Actual Events?

Our purpose in choosing a coordinate system or atlas to assign real numbers to all spacetime points is to express relationships among actual and possible events. The relationships we are interested in are order relationships (Did this event occur between those two?) and magnitude relationships (How long after A did B occur?). The date of a (point) event is the time coordinate number of the spacetime location where the event occurs. We expect all these assignments of dates to events to satisfy the requirement that event A happens before event B iff t(A) < t(B), where t(A) is the time coordinate of A. The assignments of dates to events also must satisfy the demands of our physical theories, and in this case we face serious problems involving inconsistency as when a geologist gives one date for the birth of earth and an astronomer gives a different date.

It is a big step from assigning numbers to points to assigning them to real events. Here are some of the questions that need answers. How do we determine whether a nearby event and a distant event occurred simultaneously? Assuming we want the second to be the standard unit for measuring the time interval between two events, how do we operationally define the second so we can measure whether one event occurred exactly one second later than another event? How do we know whether the clock we have is accurate? Less fundamentally, attention must also be paid to the dependency of dates due to shifting from Standard Time to Daylight Savings Time, to crossing the International Date Line, and to switching from the Julian to the Gregorian Calendar.

Let’s design a coordinate system. Suppose we have already assigned a date of zero to the event that we choose to be at the origin of our coordinate system. To assign dates to other events, we first must define a standard clock and declare that the time intervals between any two consecutive ticks of that clock are the same. The second, our conventional unit of time measurement, will be defined to be so many ticks of the standard clock. We then synchronize other clocks with the standard clock so the clocks show equal readings at the same time. The time at which a point event occurs is the number reading on the clock at rest there. If there is no clock there, the assignment process is more complicated.

We want to use clocks to assign a time even to distant events, not just to events in the immediate vicinity of the clock. To do this correctly requires some appreciation of Einstein’s theory of relativity. A major difficulty is that two nearby synchronized clocks, namely clocks that have been calibrated and set to show the same time when they are next to each other, will not in general stay synchronized if one is transported somewhere else. If they undergo the same motions and gravitational influences, they will stay synchronized; otherwise, they won’t. For more on how to assign dates to distant events, see the discussion of the relativity and conventionality of simultaneity.

As a practical matter, dates are assigned to events in a wide variety of ways. The date of the birth of the Sun is measured very differently from dates assigned to two successive crests of a light wave. For example, there are lasers whose successive crests of visible light waves pass by a given location every 10 to the minus 15 seconds. This short time isn’t measured with a stopwatch. It is computed from measurements of the light’s wavelength. We rely on electromagnetic theory for the equation connecting the periodic time of the wave to its wavelength and speed. Dates for other kinds of events also are often computed rather than directly measured with a clock. 