Hi, I'm a first year graduate student in Physics, so I probably understand string theory at just about the right level to explain the basics. If I knew any more about it, I would be smart enough to not try to explain it. If I knew any less, I couldn't explain it at all. This will all make a lot more sense if you've ever studied complex numbers. If you haven't, here's your chance to start!
First, you need to understand the geometry of regular spacetime in Einstein's Special Relativity, which isn't the Euclidean geometry with several real coordinates that you learned about in high school school. The time coordinate is a regular real variable, just like in Euclidean geometry. But the space coordinates are three different imaginary units whose square is 1, call them i, j and k. A point in spacetime is characterized by 4 coordinates, like (1t, ix, jy, kz). This system is called the hyperbolic quaternions, or Minkowski space. Why hyperbolic? Read on!
Next, how do you calculate distance in spaces with imaginary coordinates? Recall from high school geometry that in a plane with 2 real coordinates, the distance between the origin (0,0) and a point P=(1x,1y) is d^2 = x^2 + y^2 = P dot P. In imaginary coordinates you do it a little differently, you take the dot product of P with P*, P* being the complex conjugate of P, and the dot product being multiplication of only the corresponding coordinates. Complex conjugation leaves the real coordinate unchanged but flips the sign on the imaginary coordinates, so 1 goes to 1, i to -i, j to -j, k to -k. Now the distance between the origin (0,0,0,0) and a point P=(1t, ix, 0, 0) is d^2 = (1t,ix,0,0) dot (1t,-ix,-0,-0) = 1^2 t^2 + (-i)(i)x^2 = t^2 - i^2 x^2, but i^2 = 1, so we have just d^2 = t^2 - x^2. In general we have d^2 = t^2 - x^2 - y^2 - z^2. Note that different points can be distance zero from each other. These points lie on each other's "light cones" because photons travel along these zero distance trajectories. Points with positive distance from each other are called timelike with each other and can have a cause and effect relationship. Points with negative distance are called spacelike with each other and are totally disconnected.
Now we're ready to see why this geometry is called hyperbolic! What are the points which are distance 1 from the origin? Let's use the distance equation with 1 for the distance, ignoring y and z to keep the math simpler . Then 1 = t^2 - x^2, that's just a hyperbola with two branches, one in the past and one in the future! These hyperbolae go on forever and therefore so does this kind of space. This hyperbolic spacetime stuff is why objects become distorted at high relative velocities. The two spherical gold nuclei that they smash together at the relativistic heavy ion collider see each other as flat hyperboloidal pancakes.
Ok, now we're finally ready to look at these small circular dimensions. Now we use a real coordinate for time and imaginary coordinates for space, just like before. However, this time we use the normal imaginary unit whose square is -1, not 1. It's usually called i, but I've already used i, so let's just call it u. Now the distance from the origin (0,0) to a point P (1t,ux) is P dot P* = 1^2 t^2 + (u)(-u) x^2 = t^2 - u^2 x^2, but u^2 = -1, so d^2 = t^2 + x^2. The minus has become a plus! What are the points which are distance 1 from the origin? 1 = t^2 + x^2, the equation of a circle! The circumference of this unit circle gives a characteristic length to this space, usually taken to be something like the Planck Length of 1.6 x 10^-35 meters.
In string theory, spacetime becomes the product of our familiar and beloved big, hyperbolic spacetime with a bunch of these small, circular spacetimes. Particles with electric charge go around in a circle, particles with weak nuclear charge fly around on a sphere, and particles with color like quarks and gluons move around on a hypersphere. Mass is related to the size of the particle in these circular spaces, with bigger particles being lighter. When he tal
Slashdot Mirror
Hi, I'm a first year graduate student in Physics, so I probably understand string theory at just about the right level to explain the basics. If I knew any more about it, I would be smart enough to not try to explain it. If I knew any less, I couldn't explain it at all. This will all make a lot more sense if you've ever studied complex numbers. If you haven't, here's your chance to start!
First, you need to understand the geometry of regular spacetime in Einstein's Special Relativity, which isn't the Euclidean geometry with several real coordinates that you learned about in high school school. The time coordinate is a regular real variable, just like in Euclidean geometry. But the space coordinates are three different imaginary units whose square is 1, call them i, j and k. A point in spacetime is characterized by 4 coordinates, like (1t, ix, jy, kz). This system is called the hyperbolic quaternions, or Minkowski space. Why hyperbolic? Read on!
Next, how do you calculate distance in spaces with imaginary coordinates? Recall from high school geometry that in a plane with 2 real coordinates, the distance between the origin (0,0) and a point P=(1x,1y) is d^2 = x^2 + y^2 = P dot P. In imaginary coordinates you do it a little differently, you take the dot product of P with P*, P* being the complex conjugate of P, and the dot product being multiplication of only the corresponding coordinates. Complex conjugation leaves the real coordinate unchanged but flips the sign on the imaginary coordinates, so 1 goes to 1, i to -i, j to -j, k to -k. Now the distance between the origin (0,0,0,0) and a point P=(1t, ix, 0, 0) is d^2 = (1t,ix,0,0) dot (1t,-ix,-0,-0) = 1^2 t^2 + (-i)(i)x^2 = t^2 - i^2 x^2, but i^2 = 1, so we have just d^2 = t^2 - x^2. In general we have d^2 = t^2 - x^2 - y^2 - z^2. Note that different points can be distance zero from each other. These points lie on each other's "light cones" because photons travel along these zero distance trajectories. Points with positive distance from each other are called timelike with each other and can have a cause and effect relationship. Points with negative distance are called spacelike with each other and are totally disconnected.
Now we're ready to see why this geometry is called hyperbolic! What are the points which are distance 1 from the origin? Let's use the distance equation with 1 for the distance, ignoring y and z to keep the math simpler . Then 1 = t^2 - x^2, that's just a hyperbola with two branches, one in the past and one in the future! These hyperbolae go on forever and therefore so does this kind of space. This hyperbolic spacetime stuff is why objects become distorted at high relative velocities. The two spherical gold nuclei that they smash together at the relativistic heavy ion collider see each other as flat hyperboloidal pancakes.
Ok, now we're finally ready to look at these small circular dimensions. Now we use a real coordinate for time and imaginary coordinates for space, just like before. However, this time we use the normal imaginary unit whose square is -1, not 1. It's usually called i, but I've already used i, so let's just call it u. Now the distance from the origin (0,0) to a point P (1t,ux) is P dot P* = 1^2 t^2 + (u)(-u) x^2 = t^2 - u^2 x^2, but u^2 = -1, so d^2 = t^2 + x^2. The minus has become a plus! What are the points which are distance 1 from the origin? 1 = t^2 + x^2, the equation of a circle! The circumference of this unit circle gives a characteristic length to this space, usually taken to be something like the Planck Length of 1.6 x 10^-35 meters.
In string theory, spacetime becomes the product of our familiar and beloved big, hyperbolic spacetime with a bunch of these small, circular spacetimes. Particles with electric charge go around in a circle, particles with weak nuclear charge fly around on a sphere, and particles with color like quarks and gluons move around on a hypersphere. Mass is related to the size of the particle in these circular spaces, with bigger particles being lighter. When he tal