Now we need to talk about gravity and distances between branes and stuff like that.

>Is that what Randall talks about in the book?
Yes, tho' I'm making some of this up myself ... just for fun.

First, let's consider light radiating from a point source as in Figure 1. It shines on a flat brane plane and the brightest place on the plane is closest to the source of light. The light source emits, say, N photons per second. Then how many photons per second will pass through the surface of a sphere of radius R with the source at its centre? >Uh ... I'd say N per second. Good. Now, what's the area of the surface of a sphere of radius R? >I have no idea. Is it 2πR? No, it's 4πR2. That means that N photons per second pass through 4πR2 square metres. >That makes N / 4πR2 photons per square metre, right? N / 4πR2 photons per square metre per second. That's the intensity of light at a distance R from the source. Notice anything interesting? >No. It decreases with the square of the distance from the source ... just like Newton's law of gravity. We could say the density decreases ... or the probability of finding a photon decreases. Okay, now consider the intensity of light at a distance r from the brightest point on the plane. If the source is at a distance d from the plane, as in Figure 3, then the intensity at the point p is N / 4πR2. But R2 = r2 + d2 so we'd get an intensity of N / 4π (r2 + d2). Now suppose that d is very small compared to r. We'd then write the intensity as N / 4πr2(1+d2/r2). If d is really, really small compared to r, we'd get N / 4πr2, an inverse square law for gravity. >Gravity? Oops. I should mention that I'm really, really talking about gravitons, not photons ... and my plane is really a brane. >So the source of gravity is somewhere else, not on our brane?

Figure 1 Figure 2 Figure 3

Why not? Lisa talks about that in her book. The possibility of a Gravitybrane outside our own, personal brane. Remember that gravitons can travel in the Bulk, between branes. Unlike (most) other particles, they aren't confined to a brane. Our brane is the one that we know and love. It has all the elementary (and not-so-elementary) particles. Ourbrane also has that weak gravity! >But how can r be so much bigger than d? We can move as close as we please to the brightest point ... so r can be zero! Yes, but, as pointed out in the book, no measurements have ever been made on gravitational forces between masses closer than a small fraction of a millimetre.

Figure 4

When particles get too close, other forces dominate and the weak gravity forces are just background noise.
Nobuddy knows whether the inverse square law actually holds really, really close up ... like between elementary particles.
Note that both branes are 4-dimensional spacetime worlds and to get from one to the other we'd travel along a fifth dimension.

>What page are you on now?
Page 388.

Mass, Length and Time

Remember, in Part II, when we talked about the magic equation E = h c / λ = mc² ?
A measure of length is λ and m measures mass and E measures energy.
When we talk about small lengths, we're thinking: "λ is small" and that's like saying: "m is large" or "E is large".
Particles associated with small lengths are considered "massive" and require "high energy" colliders to detect.
On the other hand, when we talk about "decreasing energy" we're thinking "decreasing mass" and "increasing length".

Okay, now suppose that our distance from the Gravitybrane increases. We then think: "length increases".
So we should then be thinking: "mass decreases" and "energy decreases".

>Huh? Why?
It's a theory! Live with it!

Anyway, let's do some dimensional stuff.
Length has the dimensions L and mass M and energy has the dimensions M L² / T².

>I don't get that energy stuff.
Energy has the dimensions of mc² and c has the dimensions of (Length)/(Time) = L / T.
In fact, if Einstein had said E = mc³ you'd have been able to say: "Hey! That ain't right!"

Let's continue. If length L increases and mass M decreases how do we guarantee that energy M L² / T² also decreases?

>Don't tell me time increases as well!
You got it! As we move away from the Gravitybrane, mass and energy decrease and time increases. Neat, eh?
If there were branes closer to the Gravitybrane, masses would be smaller than in our brane and time goes more slowly (so time periods increase).
That's something like Special Relativity. Remember?

>No! As we approach the speed of light (that's "c"), mass, length, time and energy change. Remember the equations Let's consider a number of branes like Ourbrane, but positioned farther and farther from the Gravitybrane. Then each such brane-world would be just like ours ... except Mass, Length and Time would be different than ours. >Different according to who? According to whom? According to us ... assuming we could actually measure stuff on those other, similar brane-worlds.

>You're making this up, right?
Not at all. It's a neato model that's in the book.
Indeed, the model suggests that the density of gravitons (or the probability of finding one!) decreases as you move away from the Gravitybrane.
Moving away from one brane toward another would be in the direction of that fifth dimension!

>Decreases like 1/distance2. Am I right? No, it's more sophisticated than the inverse square law, but it does result in a weak gravity field on Ourbrane. >What page are you on now? Page 405.

Figure 5

Now I peek ahead in the book and see that Ourbrane is moved off to infinity!

>But that's our world, isn't it? The one with all the particles and ...
Ah, but now Lisa has another theory where we actually live on the Gravitybrane and it has all the particles and what we've called Ourbrane is now so remote it can be ignored. The result is a familiar 4-dimensional spacetime with a fifth dimension where things drop off so rapidly that we'd hardly notice the presence of that "extra" dimension ... except that it may leave traces of its presence in our world once that CERN collider gets up and running.

>And you actually understand that?
I haven't finished the book ... nor seen those extra episodes of Star Trek.