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Tag Archives: Relativity

6D invariant interval

Since one may associate either the arclength (travel length) or the arctime (travel time) with direction, one might think that the full coordinates for every event are of the form (s, t, ê), with arclength s, arctime t, and unit vector ê. Since the direction is a function of either the arclength or the arctime, the coordinates would be either (s, t, ê(s)) or (s, t, ê(t)).

However, since s = ∫ || r′(τ) || , where the integral is from 0 to t, and t = ∫ || w′(σ) || , where the integral is from 0 to s (see here), this reduces to either (t, r) or (s, w).

But science seeks unification and so must combine these forms into one. In that case, both s and t are redundant, and the full coordinates for every event are of the form [r, w]. That is, there are three dimensions of space and three dimensions of time. The arclength and arctime are implicit, and may be made explicit through integration.

The standard exposition of special relativity looks at one dimension of space and one dimension of time. This is convenient and makes Δs = Δx and Δt = Δw1. But in general Δs and Δt will either be measured directly or found through integration.

What is the distance-like invariant interval then between two events? The interval in length units (proper length) is (dσ)² = (cdw)² – (dr)²,  where c is the speed of light. The interval in time units (proper time) is (dτ)² = (dw)² – (dr/c)².

This appears different from special relativity because it substitutes the vector dw for the scalar dt. However, the scalar (dt)² = (dw1)² + (dw2)² + (dw3)² so there is no discrepancy.

In order to demonstrate that this interval is invariant for two observers traveling at different rates, one must either convert dw to dt or convert dr to ds, which reduces the six dimensions to four.

The intervals above may be generalized for general relativity with the relation L = cP √(–gμν dxμ dxν), where P is the path, gμν is the metric tensor, and there are six coordinates xμ and xν.

Ten meanings of time

Carlo Rovelli’s “Analysis of the Distinct Meanings of the Notion of “Time” in Different Physical Theories” (Il Nuovo Cimento B, Jan 1995, Vol 110, No 1, pp 81–93) describes ten distinct versions of the concept of time, which he arranges hierarchically. Here are excerpts from his article:

We find ten distinct versions of the concept of time, all used in the natural sciences, characterized by different properties, or attributes, ascribed to time. We propose a general terminology to express these differences. p.81

… our aim is to emphasize the general fact that a single, pure and sacred notion of “Time” does not exist in physics. p.82

The real line is a traditional metaphor for the idea of time. Time is frequently represented by a variable t in R. The structure of R corresponds to an ensemble of attributes that we naturally associate to the notion of time. These are the following:
a) The existence of a topology on the set of the time instants, namely the existence of a notion of two time instants being close to each other, and the characteristic “one dimensionality” of time;
b) The existence of a metric. Namely the possibility of stating that two distinct time intervals are equal in magnitude. We denote this possibility as metricity of time.
c) The existence of an ordering relation between time instants. Namely, the possibility of distinguishing the past direction from the future direction;
d) The existence of a preferred time instant, namely the present, the “now”. p.83

In the natural language, when we use the concept of time we generally assume that time is one-dimensional, metrical, external, spatially global, temporally global, unique, directed, that it implies a present, and that it allows memory and expectations. The concept of time used in Newtonian physics is one-dimensional, metrical, external, spatially global, temporally global, unique, but it is not directed and it does not have a present. In thermodynamics, time has the additional property of being directed. Proper time along world line in general relativity is one-dimensional, metrical, temporally global but it is not external, not spatially global, not unique; on the other side, the time determined by a matter clock is one-dimensional, metrical, but not temporally global, an so on. p.87

… the notion of present, of the “now” is completely absent from the description of the world in physical terms. This notion of time can be described by the structure of an affine line A. p.88

… our list does not include the possibility of considering a non-metric but directional notion of time. p.89

Table I. [without the fourth column]

Time concept Attributes Example
time of natural language memory brain
time with a present present biology
thermodynamical time directional thermodynamics
Newtonian time uniqueness Newton mechanics
special relativistic time being external special relativity
cosmological time space global proper time in cosmology
proper time time global world line proper time
clock time metricity clocks in general relativity
parameter time 1-dimensional coordinate time
no time none quantum gravity

… our hypothesis concerning time is that the concepts of time with more attributes are higher-level concepts that have no meaning at lower levels. p.91

If this hypothesis is correct, then we should deduce from it that most features of time are genuinely meaningless for general systems. p.91

… we suggest that the very notion of time, with any minimal characterization, is likely to disappear in a consistent theory that includes relativistic quantum-gravitational systems. p.91

… the concept of time, with all its attributes, is not a fundamental concept in nature, but rather that time is a progressively more specialized concept that makes sense only for progressively more special systems. p.92

Algebraic relativity

Relativity may be derived as an algebraic relation among differentials. Consider motion in the x spatial dimension, with a differential displacement, dx, differential velocity displacement, dv, and arc (elapsed) time t:

dx² = (dx/dt)²dt² = dv²dt² =  d(vt)².

Let there be a constant, c:

dx² = d(vt)² = d(cvt)²/c² = d(ct)² (v/c)² = d(ct)² (1 – (1 – v²/c²)).

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Four space and time dimensions

Since the development of relativity theory, space and time have been combined in a four-dimensional continuum. Because the speed of light is an absolute value in relativity theory, it acts as a conversion factor between space and time. Accordingly, the four dimensions may be understood as any combination of space and time:

4 + 0: Four dimensions of space and none of time. The invariant spacetime interval is commonly expressed in spatial terms only as s² = x² + y² + z² – ct²  with signature (+++–). The opposite signature is also used (+–––). The factor c converts the time coordinate into a distance coordinate. Note that there is an implicit 1-3 split in dimensions.

3 + 1: Three dimensions of space and one of time. This has been the common conception of space and time for centuries.

2 + 2: Two dimensions of space and two of time. This was discussed in the previous post here. It is a pictorial representation of four dimensions, two at a time.

1 + 3: One dimension of space and three of time. This has been discussed in many posts such as here. It may have been the ancient conception.

0 + 4: No dimension of space and four of time. This is the invariant spacetime interval expressed in temporal terms only as τ² = t² – x²/c² – y²/c² – z²/c². There is an implicit 1-3 split in dimensions.

The above should be understood in the context of an implicit six-dimensionality, as was discussed here. As soon as rates are considered, three dimensions of either space or time must be compressed to one so that only four dimensions remain.

A new geometry for space and time

This blog has described how as the distances between places cover three dimensions of space, so the durations between events cover three dimensions of time. One way of looking at this is as a map with the distance and duration given between places, such as this from the Interstate Drive Times and Distances:

There are two numbers for each leg or link in the map; one number in red for the distance in miles and another in blue for the time in hours and minutes. The durations are not proportional to the distance, which reflects the differing local conditions and topography.

It is common to adopt a speed that represents the typical speed for the mode of travel, which I’m calling the modal speed (or rate). This serves various purposes, one of which is for pre-trip planning to estimate the travel time to a destination. It can also serve as the conversion of distance and duration for the mode, much as the speed of light serves for relativity.

Divergence from the modal rate may be represented through a third dimension (z), similar to a topographic map. The modal rate then would be represented by a flat topography. A link with greater distance than the modal one would be like a positive z coordinate, and a link with less than the modal one would be like a negative z coordinate. The resultant link would be determined by the Pythagorean theorem for positive values of z, and by a hyperbolic version (as if z were imaginary) for negative values of z.

That is, |A|² = x² + y² + z² if z > 0 (Pythagorean), and |A|² = x² + y² – z² if z < 0 (hyperbolic). For positive values of z, the magnitude of the resultant link is greater than unity, that is, greater than the modal rate. But for negative values of z, the magnitude of the resultant link is less than unity, that is, less than the modal rate.

The result may be mapped like a contour or isoline map, except that negative values have a hyperbolic geometry. The simplest way to see this is to take Δt = 1 and Δr = > 1, which uses the Pythagorean theorem. If either the time interval changes to Δt < 1 or the space interval is reduced to Δr < 1, the hyperbolic version is used for the link.

Dimensions of space and time

As I’ve shown, there are three dimensions of time as well as space. That makes six dimensions in all, which I’ve written about before, such as here. There may be reasons to use the full potential six dimensions but usually it is better to contract that to four or two dimensions. We need terms to distinguish each possibility.

In classical physics and everyday life space and time are considered separate. It is common to contract 3D time into 1D time by considering only the total duration of motion. This may be expanded back to 3D time as needed.

Here are terms that refer to space and time as modeled by relativity theory:

Potential space-time (or time-space) is a six-dimensional continuum with 3D space and 3D time that models relativity theory.

Space-time is a four-dimensional continuum with space in 3D and time contracted to 1D that models relativity theory. The version associated with special relativity is called Minkowski space.

Time-space is a four-dimensional continuum with space contracted to 1D (the stance line) and time in 3D that models relativity theory. 1D space contains only the distance from a reference event.

Contracted space-time (or time-space) is a two-dimensional continuum with space and time contracted to 1D each that models relativity theory.

In classical physics and everyday use in which space and time are considered separate, the corresponding terms are as follows:

Potential space and time (or time and space) indicates 3D space and 3D time as separate continua.

Space with time indicates 3D space and 1D time displacement as separate continua.

Time with space indicates 1D displacement space and 3D time as separate continua.

Contracted space and time (or time and space) indicates space and time contracted to 1D each as separate continua.

Relativity of time at any speed

It is not well known that the Theory of Relativity is almost misnamed. Relativity was well known in physics since Galileo Galilei. That is, the relativity of space was well known. With Albert Einstein’s derivation of the Lorentz transform, the relativity of time was introduced. But the relativity of time was not of the same order as the relativity of space since it required speeds approaching that of the speed of light.

Nicholas Copernicus wrote in his 1543 book, The Revolutions of the Heavenly Spheres: “Every observed change of place is caused by a motion of either the observed object or the observer or, of course, by an unequal displacement of each. For when things move with equal speed in the same direction, the motion is not perceived, as between the observed object and the observer.”

Galileo in his 1632 book Dialogue Concerning the Two Chief World Systems noted that a person below the deck of a uniformly moving ship has no sense of movement and so objects dropped fall directly toward the feet. He articulated a principle of relativity: “any two observers moving at constant speed and direction with respect to one another will obtain the same results for all mechanical experiments.”

That is, a Galilean inertial frame of reference is in relative uniform motion to absolute space. This leads to the notion of absolute time and relative space: although different observers on different reference frames see space differently, they observe time the same way.

Einstein derived the Lorentz transformation from this foundation, allowing measures of time and space to affect one another. This solved the problem of electromagnetism and the result of the Michelson–Morley experiment.

One could say that time in the theory of relativity is relative in a minor way, whereas space is relative in a major way. The relativity of time requires speeds approaching that of the speed of light in a vacuum. The relativity of space is true at any speed.

But note one could modify Galileo’s statement to say that any two observers moving at constant pace and direction with respect to one another will obtain the same results for all mechanical experiments. In that case, time is relative at any speed, too. The key is to measure motion with pace, rather than speed, so that the independent variable is space. That allows time to be relative and have multiple dimensions.

Kinds of relativity

A simple way to look at the world is to assume that space and time are absolute: the locations, the distances, the durations, speeds, and so forth as measured by one person are the same for everyone. That is, if my automobile speedometer shows 50 mph (80 kph), then the police with a laser gun at the side of the road will show the same speed.

For many purposes of everyday life, that works just fine. But for those who think about it more or those who perform experiments, that breaks down. Galileo Galilei was the first express a principle of relativity in his 1632 work Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity on a smooth sea: any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary. He still accepted absolute time, however.

We can call Galilean relativity “spatial relativity” since it applies only to space. Since we have seen the symmetry between space and time, we could develop a similar “temporal relativity” in which time is relative but space is not. This may seem odd at first but it is as consistent (and limited) as spatial relativity. For reference, here are the transformations for spatial and temporal relativity, given two reference frames, S and S’, with an event having space and time displacements r and t (r’ and t’) respectively, with S’ moving at constant velocity v relative to S, then:

r’ = r – vt and t’ = t for spatial (Galilean) relativity, and

r’ = r and t’ = t – r/v for temporal relativity.

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Reality and relativity

As a realist I respect what is often called “common sense.” This means that our faculties of discerning reality in everyday life are basically correct. Yes, we make some mistakes, we can get fooled by a illusionist, but we almost always agree what it is that happens when things happen to us or in front of us. These are the faculties we use to determine if a mistake has been made. Our human faculties show us reality.

The corolary to this in the sciences is twofold: (1) observation is a reliable method to gather information about the universe, and (2) the observer who is closest to an object or event is the one who is in the best position to describe the object or event. This could be called the View from Everywhere (as opposed to The View from Nowhere).

One thing this means is that an observer who is at rest with respect to some object or phenomena is in the best position to measure it properly. So the proper time, length, mass, etc. of something is the measurement at rest. What other observers measure may be different but should be related to these rest measurements. That is the origin of transformations that are made between observers, or frames of reference.

At this point there are two ways to proceed: (a) either the laws of physics are defined with respect to observers at rest, or (b) the laws of physics are independent of any particular observer. Option (b) has been chosen because option (a) does not consider how observers should relate their observations to one another. So while we might say that the real time, length, mass, etc. of something is what the observer at rest measures, nevertheless it is best to transform all observations into ones that are invariant with respect to each observer.

The result is to affirm the relativity principle (RP) that Albert Einstein articulated: “all inertial [reference] frames are totally equivalent for the performance of all physical experiments.” But it also affirms that, for example, the rest mass is the real mass of a particle (not to be confused with what is called the relativistic mass).