Theoretical Science

If you read the Scientific Method then you might ask: how does theoretical science work? The answer is: they do the math!

There is an intimate relationship between theoretical physics and math. It's how a model of a physical event or process is created. Theoretical physicists create math that they think will explain a process or physical event, then they try to use the mathematical equations that they come up with to see if they can calculate observed quantities or parameters for the process or event. The idea is to create equations that will predict these events and processes.

Let's look at a typical example--Black Hole Thermodynamics. The idea here is to reconcile the laws of thermodynamics with the existence of black hole event horizons. You might ask why this is a problem! Well, it has to do with the idea of entropy. The first assumption in this problem is that black holes have entropy and therefore don't violate the second law of thermodynamics when stuff falls into the black hole. The increase in the black hole's entropy covers the decrease in entropy of the stuff that falls into it.

Let's see how they did this.

Jacob Bekenstein proposed that the black hole's entropy is proportional to the area of the event horizon divided by the Planck area. He suggested a constant of proportionality of (1/2 times ln 2)/4π.

Planck length is given as square root of (hG/C^3), where G is the gravitational constant and h is Planck's constant. This comes out to be the diameter of a proton or roughly 1.616 times 10 to the -35 power meters.

Stephen Hawking used this to come up with the Bekenstein-Hawking formula for the entropy of a black hole as S=kA/4 lp^2, where S is the entropy of the black hole, A is the area of the event horizon calculated by 4π R^2 where k is the Boltzmann's constant and lp = √ Gh/c^3 is the Plank length. Look at the image to see the final equations.

The gravitational constant G is 6.674 times 10 to -11 power N m^2/kg^2 where N is Newtons and m is meters. A Newton unit is the force needed to accelerate one kilogram of mass at the rate of one meter per second squared. This was Newton's fudge factor in his law of gravitation equation.

Boltzmann's constant k is 1.350648 times 10 to -23 Joules per Kelvin degree. This is the relation between absolute temperature (Kelvin degrees) and the kinetic energy contained in each molecule of ideal gas.

The Planck constant h is 6.626176 times 10 to the -34 power Joules-seconds. This is the energy of one quantum (photon) of electromagnetic radiation to the frequency of that radiation.

The speed of light is c, and A is the area of the event horizon.

This equation for the entropy of a black hole was not verified until 1995 when string theorists used their theories to calculate the entropy and it agrees with the Bekenstein-Hawking result.

This is how theoretical physics works out problems. The results are published in journals and then subjected to peer review. However, these theories remain theories. In the example given, no one is able to do a direct measurement on a black hole. The area of the event horizon is an estimate based on observing the black hole's effect on things near it and also from X-ray spectra and infrared scans of suspected black holes. These kinds of observations are difficult and there is some question of error.

Many theoretical physics concepts are almost impossible to prove. String theory is one of these abstract concepts. No one is able to observe strings. They're even smaller than particles, which are ridiculously tiny. The only thing that keeps these concepts going is the math, and in some cases the math is very complicated. Frankly, I'm amazed at how they came up with the math to do these things.

Thanks for reading.