Different dimensions are orthogonal to each other. In order to express a value of another dimension in a basis dimension, imaginary number is used. The values are not mixed up but the virtual distance in the mixed dimensions can be adequately calculated using imaginary number. The most typical example is the Minkowsky space. “t” is the time and “r” is the space (consisting of x, y, z space dimensions).

The Minkowsky space can be interpreted as that the time is expressed as an imaginary figure of a space dimension.

Imaginary dimension can be defined to have an imaginary order of freedom. It means that there is no chance of selecting a value but the value changes in one direction by the Tracing Dimension passing. (See the Part-2, 1. Interpretation of a time distance we observe)

The original Tracing Dimension, the time, can take any values. Therefore as a time, the time is a real number dimension having 1st order freedom. However, when it is tracing other dimensions its value varies in a single direction, which is the meaning of tracing. Therefore, the time can be regarded as an imaginary dimension of space. Of course, the space can be also described as an imaginary dimension of time.

The radius of the universe, S-dimension, is originally a space dimension for the gravitational interaction and the development of the universe. However, with the tracing by the time, the energy of the space (Spacia) takes only a single value in the S-dimension. Therefore, the S is an imaginary space dimension when the time is passing. This fact means that the S can act as a time dimension being expressed by a real number. The S is newly defined as the Time, which we observe as time going in a single direction constantly. All of the time, the S and the Time are an imaginary dimension of space.

We can conclude that

If we call only area where energy is spread as the space, the S-dimension is a time dimension by the time tracing as above mentioned. We call it as the Time. After the Big Bang, the Time is advancing. How was the Time before the Big Bang? Before the Big Bang, the energy of the universe was a boson particle (Universon) in 4 spatial dimensions. The S (= the Time) is no longer a time dimension but is a space dimension. The S dimension is not an imaginary space but a real number space dimension in the Universon. This can be also expressed that the Time (= S) is an imaginary time dimension before the Big Bang.

Prof. Hawking proposed the nature of the universe before the Bing Bang, which has an imaginary number of time. It is a 4-dimensional sphere having quantum fluctuations. This Hawking's model of the universe seems so resemble to the Bubble Universe – Time model I propose in this website in terms of time dimensions and quantum particle before the Big Bang.

The development of the universe is as follows: By the original time passing (or tracing), the universe was a quantum particle (boson), whose energy was spread in 4 space dimensions at beginning. In this stage, there was no nature of the time passing of our feeling to detect movements because energy was just a multidimensional vibration as a single quantum particle. When one of the 4 space dimensions becomes an imaginary dimension by the time tracing, it acts as a time (the Time) and the energy begins to expand extremely fast to the resting 3-D space with the Time proceeding.