Because it is well-accepted that the Standard Big Bang model cannot describe the origin of the universe, there have been many proposals which attempt to explain cosmic origins. One of these ideas which has garnered much discussion is Stephen Hawking’s “no-boundary proposal,” first popularly articulated in his 1988 best-seller A Brief History of Time. Hawking’s proposal is fairly easy to understand yet very difficult to imagine. He describes time as being finite but without a boundary. Just think of time as being analogous to a sphere: it has a finite amount of surface area but no “beginning” or “end.”
As Hawking and Mlodinow explain in their recent book The Grand Design (pg. 134-135),
In the early universe — when the universe was small enough to be governed by both general relativity and quantum theory — there were effectively four dimensions of space and none of time. … The realization that time can behave like another direction of space means that one can get rid of the problem of time having a beginning … when one combines the general theory of relativity with quantum theory, the question of what happened before the beginning of the universe is rendered meaningless.
Craig’s main criticism of this proposal focuses on the way in which Hawking converts the time dimension to a fourth spacial dimension using imaginary numbers. As Craig explained in a recent Reasonable Faith podcast,
Now, the interesting thing about this is that Hawking was able to achieve this result only by using imaginary numbers for the time variable. Now, imaginary numbers are numbers which are the products of the square root of negative one. Now, there’s no real number that is the square root of a negative number … And the problem is that although these are useful tools in computations, nobody has any idea what it would mean to talk about imaginary time anymore than talking about the imaginary volume of this room … The use of imaginary numbers is just a mathematical device to make the equations easier to solve … when you reconvert to real numbers in [Hawking’s] model, presto, the singularity reappears.
Craig then goes on to claim outright that imaginary time “has no physical significance.”
Now, it should first be pointed out that imaginary numbers aren’t any more “imaginary” than most real numbers. As mathematicians John Conway and Richard Guy write, imaginary numbers “turn out to be invaluable in many applications of mathematics to engineering, physics, and almost every other science. Moreover, these numbers obey all the rules which you already know for ‘real’ numbers” (The Book of Numbers, pg. 212).
Conway and Guy go on to explain that irrational numbers (which are a subset of “real” numbers) , such as √2 or pi, don’t truly exist in the physical sense, yet these numbers certainly go a long way in helping us to understand reality. A similar conclusion can be drawn about negative numbers. For example, does negative money make sense in the real world? Well, “negative dollars” certainly don’t exist, but they still go a long way in helping us to describe the (very real) concept of debt.
So, can imaginary numbers be used to describe the concept of time in the very early universe? Luckily, this is discussed at length in A Brief History of Time (pg. 139).
If the universe really is in such a quantum state, there would be no singularities in the history of the universe in imaginary time. … In real time, the universe has a beginning and end at singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real time is just an idea that we invent to help us describe what we think the universe is like.
Hawking then suggests that asking the question “which is real” might be irrelevant, “It is simply a matter of which is the more useful description.”
While I personally have no idea whether “imaginary time” exists or not, it seems to me that any honest person will admit that it’s at least an intriguing idea. Craig, on the other hand, seems to reject the idea of “imaginary time” outright.