I get rather irritated by the rather loose use of ‘infinity’ by physicists and science writers. At least I assume its ‘loose’ as I can see very few cases where an actual infinity can exist, other than in cases of mathematical abstractions, puzzles and paradoxes. I really assume they mean in most cases ultra tiny or extremely large, and not literally infinity as in infinitely large or small. Let’s examine the case for infinity applying to time, temperature, space or volume, density, velocity, matter and energy, and gravity.
“Ben Casey” was one of the better known and more profound of dramatic TV shows. The introduction to each episode went as follows: “Man – Woman – Birth – Death – Infinity”. Well, of course the use of the word ‘infinity’ applies only as long as man and woman get their act together, which, in all probability, won’t really be forever and ever, amen. But what you can forgive in a TV show isn’t in the same league when infinity is misused by those who should know better.
Infinity, at one end of the scale, is akin to forever and ever, amen. There is no end; there is no limit. On the other end of the scale is something more familiar to us – zero. Zero is infinitely small. So, what of the various concepts that surround us in our life, our Universe, and in our everything – can they be infinite?
Time: Time apparently comes in quantum sized bits. That is, you can’t have an interval of zero time. The smallest allowable time span, while incredibly short, isn’t zero, and is known as a Planck-Wheeler time unit. Can time be infinitely long? As long as there is change, there is time (which is rate of change), and as long as there is quantum activity in the vacuum energy there is change. That just means that if you had the ability to magnify the microscopic by many orders of magnitude, right down to the quantum level, you’d witness constant activity as virtual particles get created out of the vacuum energy and ultimately annihilate and return their borrowed energy required for their creation back to the vacuum energy.
If our Universe had a beginning, which apparently it did (the Big Bang event) and should our Universe finally stop expanding and reverse direction and ultimately all come together in the mother of all Black Holes and go ‘poof’, then clearly time, as far as our Universe is concerned, is finite, time sandwiched between the Alpha and the Omega. But, if our Universe has always been and always will be (as it would be in the ‘Steady State’ model of our Universe), or if our Universe continues to expand, ever expand, and expand some more without end, or if our Universe is one of an endless number, a cyclic or oscillating Universe that goes boom, expands, stops, contracts, goes ‘poof’, but the ‘poof’ creates another bang, etc. then time is infinitely long, even if our Universe exists for a finite segment within that infinity.
Temperature: Despite reading a comment in one of the books by the renowned astrophysicist Stephen Hawking that at the time of the Big Bang event, the temperature of the embryo Universe was infinite, you can not have an infinitely hot temperature. Temperature is the measurement of the average motion of matter’s particles. The faster they move or vibrate, the hotter things are. So, infinitely hot would imply that matter, or rather the fundamental particles that make up matter, were moving infinitely fast. But infinitely fast would mean faster than the speed of light, which is a no-no. If the Universe started off with an infinitely high temperature, then no matter how much the Universe expanded, it would remain at an infinitely high temperature, because any infinity that’s diluted, divided by a finite number, is still infinity. Since our Universe isn’t currently infinitely hot, it couldn’t have been infinitely hot at anytime in the past.
An infinitely small temperature would have to be absolute zero of course. However, absolute zero is unobtainable again because of quantum fluctuations in the vacuum energy. In physics, the lowest possible energy state isn’t ever zero because that would violate the Heisenberg Uncertainty Principle which states that it is fundamentally impossible to know both the exact location and the precise velocity simultaneously. At absolute zero, the particle wouldn’t have any motion; thus its location and its velocity would be precisely known. That’s not allowed. Since the Universe has an overall finite temperature, as it expands, it continues to cool. But, no matter how much the Universe expands, it can’t cool down to absolutely nothing.
Space or Volume: If there was a Big Bang event, a location in space and time where our Universe is defined as having a beginning, then clearly the Universe had then, has now, and will always have a finite size, albeit expanding, ever expanding. Of course, if it keeps expanding for an infinite amount of time, then after an infinite amount of time our Universe would have an infinite volume. It’s also possible that our Universe, itself finite in volume, could exist in a larger infinite sea of space. In contrast to the Big Bang scenario, a rival theory, the ‘Steady State’ type of universe where new matter is constantly being created, could be infinite in volume. As new matter is constantly being created, assuming the ‘Steady State’ universe has always been and always will be, it would need infinite space to accommodate it all. Alas, the ‘Steady State’ theory hasn’t been viable, except to a few never-say-die true believers, for decades. However, there could be a Multiverse that could, in theory, be comprised of an infinite number of individual universes, of which our Universe is but one, in which case the Multiverse space or volume would have to be infinite. A finite Multiverse (a finite number of universes) wouldn’t need an infinite amount of space, but that doesn’t preclude a finite Multiverse existing within an infinite amount of space.
Density: Density can not be infinitely small – zero – because of quantum fluctuations in the vacuum energy. Virtual particles are always popping into and out of existence in the vacuum energy. As long as you have ‘stuff’ you must have density. The concept of absolutely nothing would have zero density, but it’s not possible to achieve a state of absolutely nothing.
Density on the other end of the scale can not be infinitely large, as is frequently claimed for Black Hole singularities or the state of the Universe at the time of the Big Bang event. You’d need infinite mass to achieve infinite density and clearly neither a Black Hole, nor our Universe has infinite mass. It would also seemingly have an infinite amount of energy to expand a Universe from a singularity point of presumed infinite density, and since there isn’t an infinite amount of mass in our Universe, there can’t be an infinite amount of energy, as matter and energy are two sides of the same coin as per Einstein’s most famous equation. It’s often stated in more colourful language that mass is just frozen energy.
Either the Big Bang event singularity (and Black Hole singularities by analogy) was infinitely dense and had zero volume, or it was not. If they (Big Bang and Black Hole singularities) do not have infinite densities and zero volumes (and the alternative are really quite ridiculous), then singularities have a finite density and a finite volume and can grow in size as more stuff is added on. The upshot is that singularities can reach a size where quantum effects are negligible. Or, in other words, singularities aren’t of necessity quantum objects. To explain the importance here is that in the realm of the singularity as is currently accepted, one needs both the classical physics of gravity, and the quantum physics of the microscopic, to adequately describe the properties of any singularity. Unfortunately, there’s as of yet, no accepted theory of quantum gravity. Despite the lack of a theory of quantum gravity, you will usually read that the Big Bang event started off as an infinitely dense object of zero dimensions – a singularity. The origin of our Universe was a quantum event since the size of the singularity was within the realm of the quantum. I maintain that while as one extrapolates backwards in time towards the Big Bang event, the Universe gets smaller and smaller, it does not even reach the stage where the Universe becomes so tiny that for all practical purposes it has infinite density (that’s where the gravity comes in) and zero volume (where quantum physics rule).
Velocity: Velocity can of course appear to be infinitely small – zero – but only relative to some other object. You may be standing still relative to the Earth’s surface, but the Earth is rotating on its axis and revolving around the sun, which in turn is moving relative to other stars, etc. However, at the quantum level, those quantum fluctuations, the quantum jitters, that never ending seething activity in the vacuum energy, means that even a rock stilling apparently still on the ground is actually vibrating. In addition, the rock is expanding and contracting as the daily temperature waxes and wanes. So, appearances are deceiving here. You can’t in fact have zero velocity, and in fact having an absolute zero velocity again violates the Heisenberg Uncertainty Principle. The closest you could achieve is a thought experiment where the Universe consists of one, and only one elementary particle. There’s nothing else to measure its velocity relative to, well, something else – there is no something else.
Velocity can’t be infinitely fast as you’d have to break the light speed barrier. On the other hand, you can travel faster than the speed of light (superluminal) if that was your natural state to begin with, but hypothetical faster-than-light particles (tachyons) haven’t ever been detected, and there is no reason to believe they would travel infinitely fast even if they did exist. Anything that travelled infinitely fast in a finite Universe would be everywhere in every Planck-Wheeler unit of time. Translated, the entire Universe could consist of just one particle!
Matter and Energy: Since matter equals energy, and energy equals matter, by considering one, we consider both. Temperature is a form of energy, and we’ve already seen that temperature can neither be infinitely large nor small (zero). Further, one can’t have an infinite amount of matter without an infinite amount of space to put it in. Since we take stock in the Big Bang model of our Universe, and since that model dictates a finite volume, albeit an expanding one, it’s still finite and so the matter contained within must be finite, and therefore so is the energy content of the Universe.
And you can’t have zero matter and/or energy for reasons which should be pretty apparent by now.
Gravity: Although gravity is the weakest of the four known physical forces, it most certainly acts over long distances. Even though the pull of gravity weakens with increasing distance, it never reaches the infinitely minimum – zero. As long as there is so much as two fundamental bits of matter in the Universe, gravity will never equal zero. It’s probably nonsense to talk of gravity if there was only one elementary particle in the cosmos, since gravity is an attractive force between two or more bits of matter.
Further, because it is the amount of matter in our Universe which determines the overall gravitational quota of the Universe, and because that amount of matter is finite, that quota of gravity must be finite, even if all that matter were lumped together at one time and in one place.
Further reading in infinities:
Barrow, John D.; The Infinite Book: A Short Guide to the Boundless, Timeless and Endless; Vintage, London
Clegg, Brian; Infinity: The Quest to Think the Unthinkable; Carroll & Graf , N.Y.
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