Arnold and I chat MOND N-body simulations....

(a version of this for the humor-impaired may be found here...)

So I was sitting in my office the other day, and Arnold Schwarzenegger walked in. I was kind of busy at the time, but, you know, when Arnold walks in you sort of pay attention. The conversation went something like this:

Arnold: HEY MIHOS, there.

Me: Hey Arnold. How's California?

Arnold: Screw California. I've got bigger things on my mind. I have some questions for you. I have heard rumors that there is no dark matter, that it all can be explained by MOND. That would be good, right, we could kill off this dark matter stuff?

(<-- shh, Arnold is thinking)

Me: Well, some people think so.

Arnold: Well why don't you just run a simulation with that big fancy computer of yours? It should be child's play -- shouldn't it be just a one line change to your force law?

Me: Well, it's not that simple.

Arnold: Do it now! NOW!

Me: OK, let me explain why it's complicated. N-body simulations represent mass by individual particles, right?

Arnold: What?

Me: And we get the force on a particle by summing up the Newtonian forces from all the other particles, right?

Arnold: What?

Me: Well the standard representation of MOND acceleration as the geometric mean of the Newtonian acceleration and the MOND characteristic acceleration means that accelerations doesn't add correctly.

Arnold: What?

Me: OK, let's look at it this way. We'll take a particle with mass 4M and calculate the Newtonian acceleration a test particle at distance r would feel.

Arnold: Why 4M?

Me: hang on, just watch. So we calculate the Newtonian acceleration like this:

Arnold: Yes, I remember that from my freshman physics class.

Me: Good! OK, now lets break that particle up into 4 smaller particles each of mass M.

Arnold: I will break that puny particle with my superstrong muscles.

Me: OK, great, thanks. Now we'll arrange those particles like this:

Arnold: It looks like two crossed barbells. I must lift them.

Me: No Arnold, leave them alone. Now we again calculate the Newtonian acceleration of the test particle.

Arnold: How?

Me: Just by summing up the Newtonian forces from individual particles. Like this:

Arnold: Yah. of course, how silly of me.

Me: which we can rewrite like this:

where x = dr/r << 1

Arnold: check, please.

Me: Now we are going to expand each of these terms ...

 Arnold: You mean like my biceps expand?

Me: a series, keeping terms to x2:

Arnold: My head hurts.

Me: Maybe it's a tumor?


Me: OK, just checking. Anyway, we can simplify this with algebra. You remember algebra?

Arnold: Shut up! - no one likes a smartass.

Me: So we get

See, Arnold? We get the same answer.

Arnold: Hey, very cool, there! You computational astrophysicists are pretty smart people for punyboys.

Me: Thanks. OK, now let's do the same thing for the usual parameterization of MOND:

Arnold: That's a lot of math, there. I do not like so much math. Do not anger me, punyboy, or I will smash your grandmother into millions of tiny pieces!

Me: Hang in there, big fella. So if the original mass is 4M then the acceleration on the test particle is now:

Arnold: Can we break up that particle now? I wish to break things.

Me: Be my guest. We'll do that and add forces again, like this:

Arnold: This math looks messy. I don't like this.

Me: Don't worry, we can collect terms

and then do that series thingy again:

Arnold: Enough! STOP IT!

Me: Arnold, we're almost there. A little algebra and we get:

Do you see the problem?

Arnold: That can't be right. Maria, get the car. We're leaving.

Me: Yep, you've got it. For x<<1, we get

which is twice the original MOND acceleration.

Arnold: Mihos, you punyboy, explain it to me. Why is that a problem?

Me: Our "breaking up the particle" experiment is equivalent to running a simulation with larger particle number -- i.e., we have more particles of lower mass to represent the mass. But you've just seen that while that works under Newtonian gravity, it doesn't work in this representation of MOND. The accelerations change as you change the particle numbers.

Arnold: So you mean MOND is wrong?

Me: No, I mean implementing MOND in this way in N-body simulations -- like many people seem to be trying to do -- doesn't work. It's not a simple "one line fix" to the N-body codes. Codes that treat it that way, well, you be the judge...

Arnold: This sounds too hard for me. I will go solve California's budget problems instead. Good luck, Mihos.

Me: Good luck yourself, Arnold.

And so off Arnold went. But questions like his don't go away -- his curiosity will eventually get the best of him, and he'll be back. Yessiree, I'm sure of it -- he'll be back.