XI. Complex time and the space-time interval
Let us leave for a while the problem of the shape of the entire Universe and focus on the small fragment of space-time, in which the curvature can be ignored. In such case time can be represented by a complex number where the real part represents the local time, and the imaginary part represents the cosmic time. The real part of time may be perceived differently depending on the movement of the observer or, can be perceived as the space. And the imaginary part ensures that time flows in one direction for all observers.
How distance in such space-time can be defined? In the most simple case, assuming that the cosmic time and the local time have got the same value, then we will obtain the distance formula, as follows;
\(d = \sqrt {x^2+y^2+z^2+i^2*c^2*t^2+c^2*t^2}\),
and due to the fact that \(i^2 = -1\) it gives;
\(d = \sqrt {x^2+y^2+z^2-c^2*t^2+c^2*t^2}\),
and eventually;
\(d = \sqrt {x^2+y^2+z^2}\), . Now you can see why we can identify the distance in light-years, which is actually the distance in space-time, (because it gives the distance in space and in time), with the distance in space (the space existing only in our mind because what we see is a mixture of light from the objects that differ in age). When for example, an object is in a distance from us of 10 light-years, then we imagine that object in the direction of the reaching us light, at the distance of approximately 95 billions of kilometers. However, the object we observe is also at the distance to us of 10 years in time. If we want to have a correct perception of the Universe we must somehow determine its position in space-time. When we use the classic space-time interval we will receive zero, which provides us with no information about the real position of the object in space-time. If we estimate the distance according to our formula of the distance in space-time, we will obtain about 95 billions of kilometers, the way we imagine it.
Once again we may see that the computations and the geometrical imagination are completely different things. The result of computations of the distance in space-time is the same as in space and the additional time dimensions disappear here. However, in the geometrical model time dimensions do not disappear and we can obtain much better perception of the deployment of objects. In the geometrical model is also much easier to take into the consideration the curved space-time which will be necessary when examining the entire Universe. Time cannot be already defined by employment only a complex number because the cosmic time corresponding to an imaginary number will constitute the radius of an expanding sphere and the local time will create a spiral going towards the surface of the expanding sphere. The picture becomes more complex after adding the accelerated motion and gravitation.