__The Background__: The hands of a clock are in motion relative to the observer’s rest frame yet they display the present time of the rest frame. The motion of the clock hands is identified with the time of the rest frame.

The speed of a body is its distance traversed per unit of time. The inverse of speed is called pace, which is the time of travel per unit of length. The vector version of speed is the displacement of a body per unit of time (velocity), but what is the vector version of pace? Call it *lenticity*, which would seem to be a kind of displacement in time per unit of length, but that implies there are three dimensions of time. The rest of this article defines time and shows that it has three dimensions, although two of its dimensions are usually latent.

The term *time* has many different meanings, but it is unavoidable because of the lack of alternatives. The main thing to remember is that time in physics is *duration*, Δ*t*, an interval or length of time (cf. Newton). Let us call this the *distime* because it is analogous to distance in space. Since time is homogeneous, in physics it makes no difference what the particular date and time are. It’s the same with space: only the *length*, Δ*x*, a distance, an interval or length of space that matters, not the particular coordinates.

__The Theory__: An *observer* is a device or person capable of making measurements relative to a frame of reference. An *inertial observer* is an observer in inertial motion, i.e., one that is not accelerated with respect to an inertial system. An observer here shall mean an inertial observer. An inertial frame is one that is not accelerating. A frame of reference here shall mean an inertial frame.

An observer P measures space coordinates relative to a frame K at rest relative to P. Call frame K a *space frame* for P. *Space* (3D space) is the R^{3} geometry of places and lengths in a space frame. A *place point* is a point in 3D space. The space origin is a reference place point. The location vector of a place point is the 3D space vector to it from the space origin. The coordinates of place points are called *locations* relative to the frame K of observer P and are measured in terms of length of space. Let the space axes in K be designated as *x*_{1}, *x*_{2}, and *x*_{3}.

An observer P measures time coordinates relative to a frame L in uniform motion at a standard rate relative to P. Call frame L a *time frame* for P. *Time* (3D time) is the R^{3} geometry of times and durations in a time frame. A *time point* is a point in 3D time. The time origin is a reference time point. The chronation vector of a time point is the 3D time vector to it from the time origin. The coordinates of time points are called *chronations* relative to the frame L of observer P and are measured in terms of length of time (duration). Let the time axes of L be designated *t*_{1}, *t*_{2}, and *t*_{3}.

Every observer has a space frame and a time frame, which together form a *complete frame of reference*. Let the direction of motion of the time frame relative to the space frame be the *x*_{1} axis, and let this axis be coincident with the *t*_{1} axis. In general, the coordinates of a point in space and time will then be ((*x*_{1}, *x*_{2}, *x*_{3}); (*t*_{1}, *t*_{2}, *t*_{3})).

Since uniform motion is one-dimensional, only one coordinate from either the space frame or time frame is required as the independent denominator for rates of motion. For rates of velocity, acceleration, etc., only one time coordinate is needed; the other two time coordinates are zero, so time (duration) is a scalar. For rates of lenticity, retardation, etc., only one space coordinate is needed; the other two space coordinates are zero, so space (stance) is a scalar.

If a time frame is moving at velocity **v**_{c} relative to its associated space frame, then the other time coordinates are zero: ((*x*_{1}, *x*_{2}, *x*_{3}); (*t*_{1}, 0, 0)). The time coordinate (*t*_{1}, 0, 0) may be expressed by a scalar; call it *t*. The result is the space coordinates (*x*_{1}, *x*_{2}, *x*_{3}) and a time scalar, *t*.

Let there be a frame K_{s} with axes *x*_{1}, *x*_{2}, and *x*_{3} that is a space frame of observer P, and let there be a frame K_{t} with axes *t*_{1}, *t*_{2}, and *t*_{3} that is a time frame of P along the coincident *x*_{1}–*t*_{1} axis of motion. The velocity of K_{t} relative to K_{s} is **v**_{c}. Let event E be at rest relative to K_{s} with 6D coordinates ((*x*_{1}, *x*_{2}, *x*_{3}); (*t*_{1}, *t*_{2}, *t*_{3})). Since the time frame is moving along the coincident *x*_{1}–*t*_{1} axis, the existence of a rate of motion requires the other coordinates of either space or time to be zero.

Let there be a frame K_{t} with axes *x*_{1}´, *x*_{2}´, and *x*_{3}´ that is a space frame of observer Q, and let there be a frame K_{s} with axes *t*_{1}´, *t*_{2}´, and *t*_{3}´ that is a time frame of Q along the coincident *x*_{1}´–*t*_{1}´ axis of motion. The velocity of K_{t} relative to K_{s} is −**v**_{c} because it is in the direction opposite to **v**_{c}. Let event E be at rest in K_{t} with coordinates ((*x*_{1}, *x*_{2}, *x*_{3}); (*t*_{1}, *t*_{2}, *t*_{3})). Since the time frame is moving in the coincident *x*_{1}–*t*_{1} axis of motion, the other time coordinates are zero: ((*x*_{1}, *x*_{2}, *x*_{3}); (*t*_{1}, *t*_{2}, *t*_{3})) = ((*x*_{1}, *x*_{2}, *x*_{3}); (*t*_{1}, 0, 0)).

In this case, the frame K_{s} is a time frame, and the frame K_{t} is a space frame. This demonstrates that it is merely a matter of perspective which frame is the time frame and which frame is the space frame; physically, it makes no difference. Which is to say that space and time are a matter of perspective; they are completely interchangeable.

Let there be a frame K_{s} with axes *x*_{1}, *x*_{2}, and *x*_{3} that is a space frame of observer P, and let there be a frame K_{t} with axes *t*_{1}, *t*_{2}, and *t*_{3} that is a time frame of P along the coincident *x*_{1}–*t*_{1} axis of motion. Let there be a frame L_{s} with axes *x*_{1}´, *x*_{2}´, and *x*_{3}´ that is a space frame of observer Q, and let there be a frame L_{t} with axes *t*_{1}´, *t*_{2}´, and *t*_{3}´ that is a time frame of Q along the coincident *x*_{1}´–*t*_{1}´ axis of motion. The velocity of L_{t} relative to L_{s} is **v**_{c}. Let event E be at rest in L but moving with velocity **v** relative to K, that is, the velocity of L_{s} relative to K_{s} is **v**, and the velocity of L_{t} relative to K_{t} is *v*.

In general, the transformation of an event in K to the corresponding event in L is as follows:

*x*_{1}*´* = *x*_{1} − *vt*_{1}*,*

with the other axes unchanged from K to L. The velocity is opposite for the inverse transformation:

*x*_{1}*´* = *x*_{1} + *vt*_{1}*,*

with the other axes unchanged from K to L. This is the Galilei (Galilean) transformation.

For the time frame L, the transformation to the corresponding space frame K is similar:

*x*_{1}*´ *= *x*_{1} − *v*_{c }*t*_{1}*,*

with the other axes unchanged from K to L. The velocity is opposite for the inverse transformation.

If we take the time frame as moving with lenticity **w**_{c} rather than velocity **v**_{c} relative to its associated space frame, then it is the space frame that is linear: ((*x*_{1}, *x*_{2}, *x*_{3}); (*t*_{1}, *t*_{2}, *t*_{3})) = ((*x*_{1}, 0, 0); (*t*_{1}, *t*_{2}, *t*_{3})). Every space frame is moving at lenticity **w**_{c} relative to its associated time frame. Since the pace of |**w**_{c}| is constant, we have the time coordinates (*t*_{1}, *t*_{2}, *t*_{3})) with the space scalar *x*_{1} = *x*.

What is this space scalar? Time is linearized with a standard uniform motion in the time frame. The reading on stopwatches and clocks is associated with this linear motion in the time frame. Similarly, space is linearized a standard uniform motion in the space frame. The reading on odometers and what might be called *odologes* (odometers that do not stop) is associated with this linear motion in the space frame.

In general, the transformation of an event in K to the corresponding event in L is as follows:

*t*_{1}′ = *t*_{1} – *wx*_{1},

with the other axes unchanged from K to L. The velocity is opposite for the inverse transformation:

*t*_{1}′ = *t*_{1} + *wx*_{1},

with the other axes unchanged from K to L.

For the time frame L, the transformation to the corresponding space frame K is similar:

*t*_{1}′ = *t*_{1} – *w*_{c}*x*_{1},

with the other axes unchanged from K to L. The velocity is opposite for the inverse transformation.

If all time is local, as with stopwatches (or clocks) and time frames are disregarded, then the (3+1) Lorentz transformation follows. If instead all space is local, as with odometers (or *odologes*) and space frames are disregarded, then the (1+3) Lorentz transformation follows.

An historical example may suffice to show how space and time are interchangeable in equations. Galileo Galilei famously demonstrated the constant acceleration of falling bodies. Today this is expressed as *x* = *gt²*, where *g* is a constant. The interchange of space and time leads to *t* = *hx²*, where *h* is a constant.

This is the rate of change of time per unit of distance. The distance in this expression is *not* the distance of the falling body but the distance of a body in standard uniform motion. The time in this expression is *not* the elapsed time but the distime of a falling body in three-dimensional time.

This may be seen by differentiation and integration with zero constants:

*d*/*dt* (*gt²*) = 2*gt* ⇒ *d*/*dt* (2*gt*) = 2*g* ⇒ 2*h* ⇒ ∫ 2*h* *dx* = 2*hx* ⇒ ∫ 2*hx* *dx* = *hx*²,

where *t* is the elapsed time, *g* is a constant, *x* is the elapsed distance, and *h* is a constant. *Elapsed* indicates an independent variable.

This is not an inverse but the interchange of variables.

Galileo imagined a body moving from right to left at a constant velocity that starts falling at *b* (*Figure 4*). Galileo was the first to note that the horizontal component of the motion in space would continue moving left at a constant velocity to *c*, *d*, and *e*. But the horizontal component also provides a measure of the time for the vertical component. One can interchange both space and time measures of the horizontal component, and Galileo implicitly does exactly that.

The vertical component falls to *i*, *f*, and *h* in space and takes a semi-parabolic form. However, the vertical component also falls in time, that is, the time measured by standard uniform motion in the *vertical* dimension.

Call the axis of standard motion the space frame axis *x*_{1} and the time frame axis *t*_{1} (see *Figure 5*). Then Galileo’s switch between interpreting the sequence *abcde* as space and as time is a switch between *x*_{1} and *t*_{1}. But he could also have switched between the vertical axes *x*_{2} and *t*_{2}, that is, the fall as an extension in space and as an extension in time.

Consider this in the space frame K and the time frame L. In frame K there is a semi-parabolic motion in space. Frame L moves at the standard uniform motion that also shows a semi-parabolic motion but in time and proportional to the semi-parabola in space. In equation form:

**X**_{v}(*t*) = *gt*², and

**T**_{v}(*x*) = *hx*²,

with vertical distance **X**_{v}, elapsed distime *t*, vertical distime **X**_{v}, elapsed distance *x*, and constants *g* and *h*.