# If the universe is flat, does that imply that the Big Bang produced an infinite amount of energy?

Too much density and the universe is closed, analogous to a sphere in four dimensions: you travel in a straight line and you end up where you started. Too little and you have a saddle: not sure about the destination if you travel in a straight line. Just the right amount and the topology is flat. The flat topology is infinite: you travel in a straight line forever.

If the topology is flat (and at this point all evidence indicates that it is to within 0.4%), then multiplying the critical density by an infinite amount of cubic meters gives you an infinite energy/stress.$$\rho_{CRIT}\space kg\space m^{-3}\times \infty\space m^3=\infty\space kg$$

Is that a reasonable interpretation?

• – Qmechanic Jan 6 at 15:52
• If space is flat, then according to FLRW not only the energy of the BB was infinite, but it was infinite in every arbitrary small volume of the initially infinite universe. This is a non-physical result indicating that FLRW fails. One problem with your question is that you are asking about BB, but your formula is not for BB. In the flat case the formula for BB would be $\infty\cdot\infty=\infty$. The second problem with the question is that BB did not "require" energy, instead BB produced (created) energy. Finally, flat space is not the same as a flat universe, because spacetime is not flat. – safesphere Jan 6 at 16:35
• @safesphere - Thanks for the comment, I updated the question to make it more clear. When I said 'required', I meant, 'required' an infinite amount of energy to produce a flat topology. – Donald Airey Jan 6 at 17:40
• @DonaldAirey, what you calculate above is just the proper mass-energy of matter. It doesn't take into account the - negative - energy of the gravitational "field", which cannot be localised in General Relativity. The "total energy" of the universe could be 0, but there's no way we could give a physical sense to it, since the whole universe energy cannot be measured from "inside". I'll make this an answer. – Cham Jan 6 at 18:08
• FWIW, a succinct way to state the difference between spaces of positive, zero, or negative (constant) curvature is via Playfair's axiom, a well-known alternate expression of Euclid's 5th axiom. In a plane of 0 curvature, given a line L & a point P not on L, there's 1 line through P that doesn't intersect L (thus the lines are parallel). In a plane with +ve curvature (i.e. a sphere), there are no parallels. In a plane of -ve curvature(a hyperbolic plane), an infinite number of lines pass through P that don't intersect L. – PM 2Ring Jan 7 at 14:03

What you calculated above ($$\infty$$, for a flat space universe) is just the proper mass-energy of matter. It doesn't take into account the - negative - energy of the gravitational "field" itself, which cannot be localised in General Relativity. The "total energy" of the universe could be 0, but there's no way we could give a physical sense to it, since the whole universe energy cannot be measured from "inside".