Posted on 02/20/2015 6:01:20 PM PST by LibWhacker
Infinity Is a Beautiful Concept – And It’s Ruining Physics
I was seduced by infinity at an early age. Georg Cantor’s diagonality proof that some infinities are bigger than others mesmerized me, and his infinite hierarchy of infinities blew my mind. The assumption that something truly infinite exists in nature underlies every physics course I’ve ever taught at MIT—and, indeed, all of modern physics. But it’s an untested assumption, which begs the question: Is it actually true?
There are in fact two separate assumptions: “infinitely big” and “infinitely small.” By infinitely big, I mean that space can have infinite volume, that time can continue forever, and that there can be infinitely many physical objects. By infinitely small, I mean the continuum—the idea that even a liter of space contains an infinite number of points, that space can be stretched out indefinitely without anything bad happening, and that there are quantities in nature that can vary continuously.
The two assumptions are closely related, because inflation, the most popular explanation of our Big Bang, can create an infinite volume by stretching continuous space indefinitely. The theory of inflation has been spectacularly successful and is a leading contender for a Nobel Prize. It explains how a subatomic speck of matter transformed into a massive Big Bang, creating a huge, flat, uniform universe, with tiny density fluctuations that eventually grew into today’s galaxies and cosmic large-scale structure—all in beautiful agreement with precision measurements from experiments such as the Planck and the BICEP2 experiments. But by predicting that space isn’t just big but truly infinite, inflation has also brought about the so-called measure problem, which I view as the greatest crisis facing modern physics.
Physics is all about predicting the future from the past, but inflation seems to sabotage this. When we try to predict the probability that something particular will happen, inflation always gives the same useless answer: infinity divided by infinity. The problem is that whatever experiment you make, inflation predicts there will be infinitely many copies of you, far away in our infinite space, obtaining each physically possible outcome; and despite years of teeth-grinding in the cosmology community, no consensus has emerged on how to extract sensible answers from these infinities. So, strictly speaking, we physicists can no longer predict anything at all!
This means that today’s best theories need a major shakeup by retiring an incorrect assumption. Which one? Here’s my prime suspect: ∞.
A rubber band can’t be stretched indefinitely, because although it seems smooth and continuous, that’s merely a convenient approximation. It’s really made of atoms, and if you stretch it too far, it snaps. If we similarly retire the idea that space itself is an infinitely stretchy continuum, then a big snap of sorts stops inflation from producing an infinitely big space and the measure problem goes away. Without the infinitely small, inflation can’t make the infinitely big, so you get rid of both infinities in one fell swoop—together with many other problems plaguing modern physics, such as infinitely dense black-hole singularities and infinities popping up when we try to quantize gravity.
In the past, many venerable mathematicians were skeptical of infinity and the continuum. The legendary Carl Friedrich Gauss denied that anything infinite really exists, saying “Infinity is merely a way of speaking” and “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics.” In the past century, however, infinity has become mathematically mainstream, and most physicists and mathematicians have become so enamored with infinity that they rarely question it. Why? Basically, because infinity is an extremely convenient approximation for which we haven’t discovered convenient alternatives.
Consider, for example, the air in front of you. Keeping track of the positions and speeds of octillions of atoms would be hopelessly complicated. But if you ignore the fact that air is made of atoms and instead approximate it as a continuum—a smooth substance that has a density, pressure, and velocity at each point—you’ll find that this idealized air obeys a beautifully simple equation explaining almost everything we care about: how to build airplanes, how we hear them with sound waves, how to make weather forecasts, and so forth. Yet despite all that convenience, air of course isn’t truly continuous. I think it’s the same way for space, time, and all the other building blocks of our physical world.
Let’s face it: Despite their seductive allure, we have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable universe contains only about 1089 objects (mostly photons). If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about seventeen decimal places. Yet real numbers, with their infinitely many decimals, have infested almost every nook and cranny of physics, from the strengths of electromagnetic fields to the wave functions of quantum mechanics. We describe even a single bit of quantum information (qubit) using two real numbers involving infinitely many decimals.
Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too—in a way that’s more deep and elegant than the hacks we use for our computer simulations.
Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I’m betting that we also need to let go of it.
If f and g are differentiable in the open interval containing L [which may be a finite limit or ±∞] and if
limitx→L f'(x)/g'(x) exists,
then the indeterminate form:
limitx→L f(x)/g(x)
where f and g are both zero, or f and g are both ±∞ also exist, and
limitx→L f(x)/g(x) = limitx→L f'(x)/g'(x)
So, just for example: with f(x) = x2 g(x) = 3x2. Both are differentiable, limitx→0 f(x)/g(x) = x2/3x2 which → 0/0, an indeterminate form.
Differentiate twice: limitx→0x2/3x2 = limitx→0 2x/6x = limitx→0 2/6 = 1/3.
Obviously, you could get this answer just by "factoring out" x2. Just algebra; no Calculus required.
However, you can't factor this one: limitx→0 sin(x)/x.
L'Hospital's Rule gives:
limitx→0 sin(x)/x = limitx→0 cos(x)/1 = 1.
Remember to apply L'Hospitals Rule: you don't do the rule for differentiating a quotient.. That would give [f'(x)g(x) - g'(x)f(x)]/[g'(x)]2. You simply take f'(x)/g'(x) and check the limit.
As long as f, f', f'' and g, g', g'' [etc] are still differentiable and their quotient is indeterminate, you can apply the rule as many times as necessary to get an answer.
Can you tell me more about bijective functions, please?
Also, the fact that Zero leads to infinity, can be seen with Obama and our national debt...
Cheers!
What you said...
bttt... looking forward to reading more of this thread. :)
I think I’ll stay well outside this one.
So the upshot is that a 1-1+onto or bijective function has a unique inverse.
Thus this is a way of extending the ordinary notion of counting elements to infinite sets. When we count finite collections, we are putting them into 1-1 correspondence with a subset of the integers. To extend that notion to infinite sets, two sets have the same cardinality or "size" if there is a bijection between them.
The cardinality of the even integers is the same as the cardinality of the integers. Why? Here is a bijection f(N) = 2N.
Every non-empty open subinterval of the real line, no matter how small, has the same cardinality as the whole real line. Why? Here is a bijection: f(x) = arctan(αx); with "α" some suitable scaling factor that maps the arbitrary interval into (-π/2, π/2).
To prove the reals do not have the same cardinality as the integers, produce an enumeration of the reals, then show there is always a real number it doesn't contain. That's Cantor's Diagonalization Theorem.
Here's another way, more abstract but actually less difficult. Define the powerset of a set to be the set of all subsets of a set. So the powerset of {1, 2} is the set {{1}, {2}, {1, 2}, {}} [It's called the powerset because if a finite set has "S" elements, the set of all its subsets has 2S elements.]
Show that there is no bijection between any set and its powerset. Cantor did this already. It's the so-called "who shaves the barber" proof. Then show that there is a bijection between the reals and the powerset of the integers. Since there's a bijection between the reals and the powerset of the integers, there can't be one between the reals and the integers themselves.
In this extended sense [that there is no bijection] there are "more" reals than there are integers.
“Computer models”
Guess that solves it.
Exactly - goes hand-in-hand with Eternity. And the One who actually comprehends it all was wont to say, "I AM".
“”When I was five years old... where does Space end?””
When I was eleven or twelve (late bloomer, I suppose) I asked my Dad, “If space is continuously expanding, what is it expanding into?”
He looked at me with that same “CO form Chicago” expression, rolled his eyes at my Mom and I don’t think he answered my question either.
As a seventh grader, my speculation was that it was expanding into whipped cream. At my present age of 63 I suspect whipped cream isn’t the answer. But just being able to ask the question reminds me how “big” our Creator is and just that thought is more fun than whipped cream.
http://www.youtube.com/watch?v=iCrvibgo1LM
Thanks left that other site.
Could the answer be that the Universe isn’t expanding ‘into’ anything, it is expanding ‘from’ the start.
If there is a smallest unit of space and time and matter and energy then there is no divide by zero problem and therefore no infinity problem.
If the range of the electromagnetic force is not infinite, then it's something. What? If the lifetime of a photon is not infinite, then there are no such things as eigenstates of the Hamiltonian for electromagnetic systems. If there are no stable eigenstates, they have a lifetime. How long?
Claiming that the lifetime of a ground state is some number -- which we must discover -- is not an improvement on saying it's infinite. Its actual value might be significant of some real physics, or it may simply be an environmental value. In the latter case, we would not know that until we have searched for years or decades. Be careful what you wish for.
Max Tegmark should be reminded of this quote.
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