# How can the big bang occur mathematically?

As we know time began with the big bang. Before that there was no time, no laws, nothing. Mathematically how can an event take place when no time passes by? How did the big bang took place when there was no time?

Note my question is not about weather big bang took place or not, my question is about is this a mathematical anomaly? Thanks

-
Of course, we don't know whether laws or time existed at the instant..! But, we can visualize what can there be from the next second or so... ;-) –  Waffle's Crazy Peanut May 22 '13 at 13:53
–  dmckee May 22 '13 at 13:54

I can't improve on Phil's answer, but let me present a slightly different perspective.

The name "Big Bang" originally referred to the time zero limit of the FLRW metric. Since Friedmann et al inflation and quantum gravity have muddied the waters, but let's stick with the FLRW description and ignore the complications.

If you make a few simplifying assumptions about the universe you can solve the field equation and obtain the FLRW metric. Once we have a metric we can examine the evolution of the universe with time, and in particular work backwards to time zero. If we follow time back to zero we find the curvature goes to infinity, and this happens in a finite proper time (13.7 billion years in fact). Since we can't do arithmetic with infinity we can't integrate further back to times less than zero, so it's commonly said that time started at the Big Bang. Hence your confusion about what preceded it, and your statement:

Before that there was no time, no laws, nothing.

But actually you can't make the above statement. All that can be said for certain is that GR has no way of describing what happened before the Big Bang. This is quite different to the statement that there was nothing before the Big Bang because we know GR can only be an approximate theory at very small distances and very high curvatures.

As Phil mentioned, it's widely believed that near (apparent) singularities we can only get a good description of the physics by using some theory of quantum gravity. We expect this will allow trajectories to be calculated back through the Big Bang to times preceding it. The proponents of Loop Quantum Cosmology already claim they can describe what happens at the Big Bang, and they find it's actually a Big Bounce and the singularity never forms. However these calculations are far from universally accepted. I believe some proponents of String Cosmology have also claimed to predict a bounce, but I don't know a lot about this area and I believe the claims are rather speculative.

-

We do not have a complete mathematical description of the big bang. That would require a theory of quantum gravity because effects of both gravity and quantum mechanics become relevant when you want to describe what happens under the physical conditions that we think were important in the big bang. Quantum Gravity is still a subject of active research. However, there is no mathematical difficulty in principle to describe a universe where time starts at the big bang.

In mathematics we can construct objects using a suitable set of axioms such as those of set theory which we assume to be consistent. The positive real numbers are a well defined set and the set of functions from the positive real numbers to possible states of the universe in some theory can also be a well defined set. This means that there is no mathematical obstacle to describing a universe that starts at time zero. That does not mean that this is the right way to describe the big bang. It is just an example to show that there is no mathematical problem.

Perhaps the problem that you are really thinking of is causality. The principle of causality says that every effect has a preceding cause. This is not a mathematical axiom. It is a physical assumption based on philosophical considerations that may or may not be right. If causality is absolutely correct then there would need to be something before the big bang to make it happen. Many physicists and philosophers assume that this is the case and they construct theories of the universe where the big bang is not the beginning. This includes cyclic universes of various sorts, eternal inflation and evolutionary models with baby universes. All these theories are mathematically consistent but are based on pure speculation. Anyone could be right or they may all be wrong.

There are other physicists who do not think that causality is always applicable in which case it is not necessary to have anything before the big bang. In fact causality is a consequence of the second law of dynamics which is purely a statistical feature of systems with a large number of degrees of freedom. At the atomic or sub-atomic level all physics is reversible and there is no way to distinguish cause and effect from each other. The classical laws of physics can be written as a principle of least action which does indicate any difference between initial and final conditions. In quantum theory this becomes a path integral over all kinematically valid ways the universe can evolve. If this includes space-time geometry as it should in quantum grvaity, then the beginning of the universe can be described by a sum over space-times that smoothly curve back on themselves at the big bang. This is the Hawking-Hartle theory of the big bang. It is probably not fully correct but it does show that quite reasonable mathematical descriptions of the big bang with no causality are perfectly consistent and scientific.

-
"If causality is absolutely correct then there would need to be something before the big bang to make it happen." Not true. In classical GR, we have a universe at all times $t>0$. For every time $t$, there is another time $0<t'<t$ that could be considered to have caused $t$. The way GR captures causality is with the notion of a globally hyperbolic spacetime (Hawking and Ellis, p. 206). Realistic cosmological models using GR are globally hyperbolic. –  Ben Crowell May 23 '13 at 0:26
Ben, by the same argument I can say that everything that happened since noon yesterday was caused by events just after noon and causality does not imply that anything happened before noon. Such arguments just make nonsense out of the notion of causality and if you want someone to defend that you need to put it to someone who thinks causality is fundamental, i.e. not me. –  Philip Gibbs - inactive May 23 '13 at 6:36

You asked for a purely mathematical answer and received a correct physical answer from John Rennie. If you want a purely mathematical answer, that purely mathematical answer has to exist within a well-defined mathematical theory. As John has explained, we don't have a well-defined mathematical theory of the Big Bang that is also an acceptable, complete physical theory. We do, however, have one pretty serviceable, well-defined mathematical theory of the Big Bang, which is the one provided by general relativity (GR).

As we know time began with the big bang. Before that there was no time, no laws, nothing. Mathematically how can an event take place when no time passes by? How did the big bang took place when there was no time?

In the context of GR, the answer to this is extremely simple. An event in GR is a point in the manifold that represents spacetime. The Big Bang is a singularity, and singularities are not points on the manifold. Therefore the Big Bang is not defined mathematically as an event (or set of events).

-
Why the downvote? –  Ben Crowell May 23 '13 at 15:02

According to dictionary.com:

An event is an occurrence that is sharply localized at a single point in space and instant of time.

An event can't take place without time. Yet, many scientists claim that there was no time before the big bang. But, that is completely illogical. How can there be no time and then suddenly an event take place which requires time. That is circle (or paradoxical) reasoning . . . something that many scientists use when they don't know what to say.

-