# On Gravity and Levity

Welcome to Gravity and Levity, a blog devoted (mostly) to a conceptual discussion of the big ideas in physics.

Gravity and Levity is not intended to be a resource for the questions “What is the solution to/equation for ____ ?” or “How do you solve _____ ?”. These sorts of questions are, for the most part, easily answered using Wikipedia or the various physics forums. Rather, the purpose of this blog is to tackle the much more nebulous and difficult “How do you think about ____?”. I firmly believe that behind every equation in physics is a conceptual and often crazy idea. I created this blog to explain what little I know about these big ideas and to get others to discuss them.

So here’s what you can expect to find on Gravity and Levity:

- My attempt to describe the “picture” behind some of the more confusing ideas in upper level physics: the principle of least action, quantum bound states, entropy and free energy, functional integration, universality of the speed of light, Friedel oscillations, the quantum field, etc.
- Lots of cheap, pencil-and-paper drawings.
- Random “fun” posts, wherein I discuss interesting or amusing non-physics ideas that have been on my mind. Topics may range from crazy facts about wildlife to professional basketball to my own thoughts about how we perceive time as we age. Be warned.
- Weekly updates. [Update: Okay, so this prediction turned out to be a little too optimistic.]

Here’s what you should *not* expect to find:

- Exact solutions. This is a forum for discussing ideas, not a resource for homework help.
- Equations. Some mathematics is unavoidable, but I almost always prefer a good picture to an equation.
- Current events and personalities in science. I’m not going to spend much time talking about famous scientists or the status of big experiments. The politics/news/gossip of physics is extremely uninteresting to me.
- “Science vs. Religion” rants. All of the popular science forums are dominated by this kind of babble, and it gets tiring really quickly. Besides, the Wikipedia article does a surprisingly good job with it.

Comments and suggestions for future posts are always welcome.

The author of this blog, Brian Skinner, is a postdoc in theoretical condensed matter physics who hopes to make a career in physics teaching and research. The story/rationale behind this blog’s creation is explained in the blog’s first post, and a short introduction to the author is given here.

The banner image is taken from wikimedia commons.

I’ve really enjoyed your posts. I’d like to subscribe by RSS feed. Do you have one for the posts, or did you choose not to have one? I apologize if I have just overlooked its location.

I have to admit that I am pretty ignorant about the workings of the blogosphere, and I’m just using the free version of wordpress so my ability to customize is quite limited. But I think that you can subscribe here: http://gravityandlevity.wordpress.com/feed/rss/

If that doesn’t answer your question, there’s more information about wordpress feeds here: http://codex.wordpress.org/WordPress_Feeds ; maybe you can understand how this blog works better than I can. : )

Dear G & L — as one who has loved physics since my teen years, but discovered in college that I just-couldn’t-do it, I’m thrilled to discover this site. I try to keep up my attempts at understanding modern physics as best I can but the concepts are slippery. I may not understand much, but I’ve never been happy with the physics stance that some things can’t ever be understood: for ex., we don’t understand wave/particle duality because we “can’t.” In my HUMBLE opinion that’s baloney. The truth under quantum physics may be baffling to us, but there is SOMETHING going on.

In high school, I would always try to tie the equations I learned back to the reality I was trying to describe. And this seems to be the purpose of this blog — at a slightly more sophisticated level. So thank you. There have been many times I’ve re-read and re-read an article on physics trying to decipher what the heck it actually means, and I think you are going to help a lot on that score. I look forward to reading the rest of the blog entries.

Mark

I just had a couple of ideas in the middle of the night and I was looking for someone to bounce them off of and happened to find this place. I’m curious to know if any of them are plausibly workable.

I was thinking about the wave function of the universe. It starts out at time zero with the big bang, a state where the amplitude of the wave function is infinitely dense. Over time the wave function spreads out until after an infinite amount of time it approaches zero density over all space.

So I started thinking about if you could define a mapping between state vectors of the universe so that the dense states are mapped to sparse states and vice versa. The idea I’m looking for is to do this in such a way so that the time reversal of the universe is really just the same thing as forwards, but in a wackier way than normal time symmetry.

So I was thinking you could do this by simply inverting all of the amplitudes. But the problem with this is that the big bang is infinitely dense in an infinitely small area and the other end is infinitely sparse in an infinitely large area and so if you invert the amplitudes of the infinitely sparse state, you’ll get something that has to be renormalized in order for the amplitudes to integrate to 1.

So then I started thinking, what if the big bang state is infinitely dense over infinite space and then you normalize it shrinking space. As it gets less dense, you stretch the space out to keep it normalized. This also conveniently gives the expansion of the universe.

Does any of that make sense? You won’t hurt my feelings if you say no.

Hi Matt,

What you’re looking for is called a “symmetry transformation”, and people in physics talk about those all the time. So in that sense the question your asking is completely sensible.

I don’t think that something as simple as “inverting the amplitudes” will work, though. Try testing it out on a simple test case, like the wave function for a free particle that starts with a definite position and a finite range \delta_p of momentum states. I think that wavefunction should look like this:

P(x,t) = 1 / (\hbar^2 \delta_p^2 pi t)^(1/4) * e^(-x^2/(2 \hbar^2 \delta_p^2 t))

This function is a delta function at t = 0 (perfectly dense around x = 0) and is a gaussian with increasing width at later times.

I don’t know that this is an adequate description of the wavefunction of the universe, but it seems reasonable to me. You can try out different transformations to see whether something will give you a result P(x,t) —(x,t) get transformed to (x’,t’)—> P(x’,t’). i.e. the form of the wavefunction should be the same after your transformation.

Finding these kind of symmetry transformations can be a very big deal. They can tell you what is constant (conserved) in a system, which allows you to make definite predictions about the future.

I think what I’m looking for is a way to interpret the wave function “inside-out” and flip-flop the concepts of density and sparsity.

I’ve been playing around and I think I’ve found what I’m after. Let P(x,t) be defined as above (although I think it should be pi^2 rather than pi). Now I define P’(x,t)=1/(\hbar^2 \delta_p^2 pi^2 1/t)^(1/4) * e^(-t/(2 \hbar^2 \delta_p^2 x^2)).

Note the following relationship between P and P’:

P’(x,t)= 1/P(i x, t)/(pi \hbar \sqrt(t) \delta_p)

P(x,t)=1/P’(i x, t)/(pi \hbar \sqrt(t) \delta_p)

What I’m trying to do here is make P’ represent the chance of not finding a particle in a particular place. When I integrate P(x,t)^2 over the interval (-a,a) I get Erf(a/(\hbar pi^1/2 t^1/2 delta_p)). When I sum the integrals of P’(x,t)^2 over the intervals (-infinity, -a) and (a, infinity) I get 1-Erf(a/(\hbar pi^1/2 t^1/2 \delta_p)). Good, this means when you integrate P’ over the complement of an interval, you get the complement of the probability that the particle is not in the original interval.

Now the final bit piece. Note that the following relationship holds:

P(x,t)=P’(1/x,1/t)

So as I’m interpreting this, when you map the end of time onto the beginning of time and the edge of the universe onto the origin, then you get the same thing as the normal wave function, only the interpretation of the wave function is flipped so that finding a particle becomes not finding a particle and vice versa.

Crap, scratch that thought. Looks like all I managed to do was make Mathematica do something stupid.

You can try something like an inversion transform: http://en.wikipedia.org/wiki/Method_of_inversion#Method_of_Inversion, although I expect it would only work on a zero energy wavefunction.

By the way, you might enjoy this interesting and highly readable paper that discusses the wavefunction of the universe, how simple we can expect it to be, and what symmetries it might have:

http://space.mit.edu/home/tegmark/nihilo.ps

(or here: http://www.springerlink.com/content/u4v7223rnh8h6414/ )

Thanks for the link. I was familiar with some of those ideas but not all of them.

The direction I’m thinking about now, is what if you could define the wave function without the usual constraint that the integral of the square of the amplitude has to converge to 1 and then specify a normalization procedure which stretches space in a manner similar to general relativity to make it converge to 1.

The obvious problem with this idea is that it would imply that distributions of matter in other branches of the wave function far different from our branch would have be influencing the structure of space-time in our branch, which wouldn’t match up with general relativity.

So my thought is to take a step back and think if it’s possible to give every branch it’s own space-time that relates smoothly with the structure of other branches. Would it be possible to have a metric that combines both distance in space-time and a measure of similarity of states such as trace distance?

I’m not sure if this makes sense, but what I’m thinking is, if our space-time was influenced gravitationally by alternate earths slightly displaced from our position, they would all net out and yield something that wouldn’t be distinguishable from ordinary gravity. And under this line of thinking, configurations greatly dissimilar from our own would only have a small impact on us due to a large dissimilarity in the distribution of matter.

So all of these vastly different distributions of matter would still “interact” (although not in the same sense as normal physical interactions) with us, but the effect would be small and it would be roughly even in all directions. Wouldn’t that be equivalent to the cosmological constant in relativity?

Another thought is that if you assume that everything self-normalizes by stretching space-time, then couldn’t that resolve the non-convergence problems in QED?

Of course I’m pretty sure all of this requires breaking some (most? all?) of the postulates of QM and GR, but I wonder how much they could be altered without substantially altering the predictions of the theories.

This might be the wrong place, but I can’t find a link to subscribe. I would suggest you add that widget. I could be blind though.

Cool blog!

Okay, I think I just added a subscribe/follow widget. Let me know if it doesn’t work for you!