4.2 Gravity

summary

How do we connect quantum mechanics with relativity?

We have two kinds of theories in nature—the very small is described by quantum mechanics, quantum field theory, and particle physics; general relativity describes the very large.
Yet there is one “nature” and so they should connect. We need something to “glue” the ends together—something called quantum gravity.
When we think about gravity in the scope of general relativity, we see that spacetime’s curvature is affected by massive objects. The way other objects move in this curvature of spacetime is how we perceive gravity.
This is very different from how we think about the other forces, as interactions of particles. Gravity is instead seen as ripples in spacetime.
These ripples form waves, which we see indirectly in measurements. As you “dial down” the energy of these ripples, at some point you encounter a quantum of energy—a massless, spin-2 particle whose properties can be computed. This is the graviton.
Supersymmetry allows for a spin-2 particle, but only one.

Making the connection to quantum mechanics allows for new ways of studying gravity.

Just like any other particle-particle interaction, we can look at the scattering of two gravitons.
The strength of the graviton’s self-interaction can be calculated—this number is Newton’s constant.
Max Planck devised a way to look at all physical properties in natural units, specifically by using the Planck length and Planck mass:$$l_{\textrm{Planck}}=\sqrt{\frac{\hbar G_N}{c^3}}\sim 10^{-35}\,\textrm{m} ~~~~~~~~~~~~~~~~ M_{\textrm{Planck}}=\sqrt{\frac{\hbar c}{G_N}}\sim 10^{19}\,\textrm{GeV}$$ and so we can say that$$G_N\sim\frac{1}{M_{\textrm{Planck}}^2}$$where $G_N$ is Newton’s constant.
Mysteriously, the force of gravity is much weaker than the other three fundamental forces. Compared to the electric force, gravity is nearly negligible. $$F_{\textrm{grav}}=\frac{m_1m_2}{M_{\textrm{Planck}}^2}\frac{1}{r^2}$$
Quantum mechanics gives ripples in everything—in particular, fields—so it should also ripple spacetime itself. There should be a natural unit where spacetime becomes quantum foam.
In a sense, you could keep magnifying spacetime, and eventually see these sort of “atoms” of spacetime itself, the size of the Planck length.

String theory provides a new mathematical way to describe gravity.

The self-interactions for gravitons contain a mathematical inconsistency—as you start to compute the virtual particles, adding more and more of these particles actually results in the effect becoming stronger and stronger, when we should expect it to become weaker.
A similar problem was solved in the theory of radioactive decay of particles. For instance, a proton decaying into a neutron was initially thought of as a point-interaction. But this, too, was mathematically inconsistent with the Standard Model. This was solved with the mediation of a force-carrying particle of known mass, specifically the W boson.
So perhaps there is a similar intermediate string on the order of the Planck mass in the graviton-graviton scattering scenario.
A string can oscillate and vibrate, and the lowest order of vibration produces a spin-2 particle. You can think of a string as a “dressed up” graviton—a graviton together with many other particles.
Looking at string interactions rather than particle interactions provides a smooth and mathematically consistent theory of quantum gravity; in fact, string theory is the only method by which we have managed to describe quantum gravity.

Einstein's fundamental idea of unification was to explain physical phenomena using geometry.

Kaluza-Klein theory was developed in the 1920s, built around the idea of a fifth physical dimension outside the usual four (space + time). The fourth spatial dimension was “curled up” such that a particle moving in this fifth dimension would be moving in a circle around the surface of a cylinder.
Under the laws of quantum mechanics, the speed at which the particle could move in this circular fashion was quantized. The physical manifestation of this quantized speed was actually a description of a particle with an electric charge.
If you take gravity in five spacetime dimensions and reduce it back down to four dimensions mathematically, you recover gravity along with electromagnetism.
However, if you were to compute the size of this extra fifth dimension, you would discover that it is on the order of the Planck length. Thus, this unified theory can only work if we consider quantum gravity.
Einstein grappled with the laws of quantum mechanics for much of his life, constantly trying to find inconsistencies and paradoxes. He believed quantum mechanics was correct, but incomplete. Now in modern physics, we see this idea of explaining the infinitely large together with the infinitesimally small. By using geometrical tools in string theory, Einstein’s idea of unifying physical phenomena with geometry still lives on today.