Space and time are joined together to make space-time in Einsteinian relativity. Space-time suggests a new concept of distance, and in Einsteinian relativity, this new distance is called Minkowski’s space-time Interval (S):

E 6.6 (In Cartesian coordinates x,y,z,t)

Surprisingly, Minkowski’s interval (distance in space-time) may be real, zero, or even imaginary between any two different events in space-time. Simply, the surface of light cone (the hyper-cone that is formed after a flash of light is emitted) represents the null interval in space-time, although the light cone is not a single point in space-time.

Additionally, the value of Minkowski’s interval between two events in space-time depends on the orientation of the coordinate system, and it implies a direction in space-time (before after).

These consequences are the result of Minkowski’s assumption: He assumes time as a dissimilar quantity, and accepts the sign of “time” as negative in the Pythagorean equation.

We will not discuss Minkowski’s space-time interval with all details here, but please note that Minkowski’s assumption is a result of considering space-time as static. He assumes that the surface of light cone is a null interval. However, the space-time geometry that we suggest here fundamentally expands (Section 4.2), and we assume that the light cone is the expansion of a local space (oblique radial) in the expanding space-time geometry, where the expansion is strained universally (Section 5.4).

In general, distance is a numerical description of how far apart things lie, and it can be positive between two different points, and is zero precisely from a point to itself. The concept of distance does not contain information about direction, since it is a scalar quantity, and it does not depend on the orientation of the coordinate system.

Now, we will suggest another formulation of distance in space-time, which is more appropriate with regard to the general definition of distance. Contrary to Minkowski’s interval, we suggest to formulate four-dimensional (space-time) distance interval by including distance towards time dimension in the Pythagorean equation with a positive sign.

Definition: Distance in space-time (D4d)

D4d is a four dimensional distance interval in space-time.

or

E 6.7 (In Cartesian coordinates)

We can include the concept of time into the Pythagorean equation with a positive sign; since, we clearly distinguished different meanings of the concept of time: Time as quantity of clock-ticks (Section 6.3) and distance towards time dimension (Section 6.2). Interestingly, as we discussed, distance towards time dimension is exactly a geometric quantity, which is not a relative property of matter like timeas clock-ticks. Distance towards time dimension is the difference of the radius of curvatures of two points in space-time:

Figure 6.11 Distance toward time dimension

Distance in space-time (D4d) is also a property of space-time geometry like distance towards time dimension (Dtd), while time as quantity of clock-ticks is the property of each individual knot or vortex.

Please note that all definitions of distance in space-time (D4d) are based on the degree of radial obliquity angle (θradial obliquity). However, we will not tackle the issue of determining the exact numeric value of radial obliquity angle until further chapters, because it cannot be measured by direct observations, but it can be deduced by analyzing the nature of fundamental forces.

Now, let us discuss the physical meaning of simultaneity.

It is not possible to map the entire universe by using Cartesian coordinates, since space-time has curvature. The formulations above should be accepted as symbolic expressions.

The degree of radial obliquity angle (θ radial obliquity) varies in gravitational fields. In fact, we will formulate relative metric contraction in gravitational fields as a function of the obliquity angle.