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Relativity, Cosmology, and Time

Euclidean Space

The Minkowski space-time in relativity has the signature (-,+,+,+). The negative signature in the time component has obscure properties that make visualization very difficult as well as creating novel features not easily comprehensible. In many instances, the problem can be resolved partially by substituting t it, which transforms the Minkowski space-time to the Euclidean four-dimensional space with signature (+,+,+,+). Sometimes, the transformation is reversed back to the original Minkowski space-time at the end of the computation. However, some physicists prefer to treat the Euclidean space as the ultimate reality, while the Minkowski space-time is considered just a figment of our imagination - a point of view not yet validated by observation. Actually, even a four-dimensional Euclidean space is very difficult to visualize. So let us start with a three-dimensional space as we experienced all our life.

3-D Sphere In 3-D Euclidean space, the formula for a sphere of radius a located at (0,0,0) has the form:

x2 + y2 + z2 = a2

with the metric ds2 = dx2 + dy2 + dz2 = a2 (d2 + sin2 d2) ---------- (20g)

in spherical coordinates (r, , ), where the radius r = a is constant (Figure 10p). The sphere has only two degrees of freedom. Further simplification is possible by considering only those circles around the z-axis, or z = a cos = constant, which implies = constant. In this case

Figure 10p 2-D Sphere
[large image]


ds2 = a2 sin2 d2. The length of the circumference is 2 a sin, which varies from 0 to a maximum of 2a and falling back to 0 as varies from 0 to /2 and then to . The point at
= 0 or (sin = 0 at the North and South poles) is not a singularity as it can always be removed by re-defining the origin of reference.


In the four-dimensional Euclidean space, the 3-D hypersphere is defined by the formula:

x12 + x22 + x32 + x42 = a2

and ds2 = dx12 + dx22 + dx32 + dx42 = a2 (d2 + sin2 d22) ---------- (20h).

where d22 = (d2 + sin2 d2) is an unit 2-D sphere equivalent to d2, and d2 assumes the role of d2 in Eq.(20g).

The formulation can be generalized further to 5-D Euclidean space with the 4-D hypersphere metric:

ds2 = a2 (d2 + sin2 d32) ---------- (20i).

where the 3-D d32 = (d2 + sin2 d22) is equivalent to d22 in Eq.(20h).
Hybrid Universe The no boundary proposal makes use of Eq.(20i) and suggests that at the beginning the universe was running with as the time component from = 0 to /2 in the Euclidean space. At /2 the character of the time component underwent a transformation, e.g., /2 + ict/a. Since then the 4-D hypersphere has changed into a Minkowski hyperboloid described by the metric:
ds2 = -c2dt2 + a2 [cosh2(ct/a) d32]
where the 3-D hypersphere is expanding as "a cosh(ct/a)", and approaches a de Sitter universe in the form (a/2)ect/a as ct/a +. Such a hybrid universe is shown in Figure 10q. Thus, there is no singularity at the beginning of time. That particular region becomes as smooth as the North

Figure 10q Hybrid Uni-verse [view large image]

(South) pole on Earth. It is suggested that the (red) region in Euclidean time represents the moment of nucleation via tunneling through some sort of energy barrier from nothing (in the form of vacuum fluctuation for example).
The expression for d22 is just the metric on 2-D unit sphere (an ordinary sphere with radius equal to 1) in a 3-D space as shown by Eq.(20g). Similarly, d32 is the metric on 3-D unit hypersphere in a 4-D space. In general, dn2 denotes the metric on n-dimensional hypersphere in a (n+1)-dimensional space. It is always possible to create n-dimensional hypersphere and other geometrical shapes in a (n+1) dimensional space including the Euclidean and many others. For example, a 4-D hypersphere can be embedded in a 5-D anti-de Sitter space with metric in the following form:

ds2 = d2 + sinh2() d42 ---------- (20j).

where d42 = (d2 + sin2 d32) is the 4-D hypersphere.

This 5-D AdS space has been used to apply the holographic principle to physical theories or objects (such as superstring theory or black hole) by encoding them from such 5-D space to a 4-D hypersphere.

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