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) || dτ, where the integral is from 0 to t, and t = ∫ || w′(σ) || dσ, 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 = c ∫_P √(–g_μν dx^μ dx^ν), where P is the path, g_μν is the metric tensor, and there are six coordinates x^μ and x^ν.

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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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.

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.

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.

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).