According to the above video, an ultra-diffuse galaxy (far fewer stars than ours) contains little or no dark matter. Light passes through it in a seemingly straight line (little or no gravitational lensing). Then there's the dark matter galaxy, with far fewer stars than our galaxy. Most galaxies appear to have approximately five to ten times more dark matter than baryonic matter.
Using Einstein's field equations, we shall build these different types of galaxies and derive an equation that models each of them. However, to accomplish this task, we make the following assumptions: A dark matter particle is an excitation of a field, possibly dark energy or the vacuum. Like the vacuum of space, dark matter particles are diffuse. They have no electromagnetic force. They don't interact significantly with ordinary-matter particles or with themselves. There is, however, gravitational attraction. Finally, dark matter has something similar to the Pauli exclusion principle which prevents DM particles from gravitationally collapsing into a single point in space.
Because dark matter is diffuse, its gravity is insignificant near, say, a black hole, but becomes more significant as we increase the distance (r). Below is a crude illustration of a dark-matter galaxy. It has a black hole in the center. Inside the smaller circle, there's the big black dot (black hole) surrounded by only a few small dots (dark matter). By contrast, inside the large circle the small dots dominate (make up most of the mass).
Now that we've defined "dark matter," let's define the variables we will need:
We are ready to begin working with Einstein's field equations:
Let's change the cosmological constant term and make a substitution:
We make the substitution at equation 3. At equation 4 we put the term on the other side of the equals sign:
To make the math less clunky, let's change the terms into some simple variables:
After making more substitutions we get equation 12 which has a surprising implication!
Spacetime can be flat no matter how much mass (energy density, pressure, etc.) you have! Variables C and D are the energy-stress or mass terms. As long as they are equal, spacetime is flat--which implies no gravity! How can this be?
If there is no gravity between C and D, they will not move (or they will be pushed apart by dark energy). Imagine a red bowling ball and a blue one sitting apart from each other on a trampoline. Imagine the trampoline is expanding. It might look something like this:
Each bowling ball makes a dent in the trampoline, but they don't roll to each other. But around the region of each dent, there are some marbles that roll toward each bowling ball. The analogy demonstrates the following: Gravity is local and not global. If two or more masses have equal energy densities, there is no attraction. Where they are not equal, there is attraction.
The next diagram shows why galaxies move away from each other rather than towards each other:
The above diagram is a toy universe consisting of four multi-colored galaxies. They have equal masses and are the same distance apart. The arrows assume there is gravity between them, but the red galaxy is equally attracted to the green and blue galaxies. The other galaxies are in a similar predicament. The gravity of one galaxy is cancelled by the gravity of another galaxy. So there's no movement. This universe is static. Even though there is plenty of mass, spacetime may as well be flat and gravity may as well be zero. Add some dark energy and this universe will expand indefinitely.
So how do we get some gravity going in our toy universe? We break the symmetry. One way is to alter the distances between the galaxies (see first diagram below). Another way is to vary the masses (see second diagram below).
Now, you may be thinking, this is all fine and good, but what does this have to do with dark matter? Well, to understand dark matter, we have to have a better understanding of how gravity works and how it doesn't work. The main point you want to hang on to is equal masses that are equidistant aren't attracted to each other or that attraction is cancelled. Thus galaxies stay apart and move apart except when the symmetry is broken:
Now, what is true for massive galaxies must also be true for smaller stuff like dark matter/energy particles:
So all we have to do to make a dark matter galaxy or any galaxy containing dark matter is break the symmetry by introducing, say, a huge mass:
Let's assume the big mass is baryonic matter. Why baryonic matter? Baryonic matter is electromagnetic. It can form into molecules, planets and stars. It can come in large clumps. Whereas dark matter is non-electromagnetic and does not form clumps. It remains diffuse much like dark energy or the vacuum of space. When its particles are equally spaced, they are not gravitationally attracted to each other, so they need a catalyst, a symmetry breaker. Baryonic matter comes in different-sized clumps and is a perfect symmetry breaker.
In our toy universe we can imagine little dark matter particles falling into the spacetime dent created by the huge clump of baryonic matter. The dent, of course, gets deeper, allowing even more DM particles to enter:
So when does this process end? It looks like it could go on indefinitely! Dark matter pours into the dent, the dent gets deeper, more particles pour in, and so on. There must be an equilibrium where the mass of the dark matter is around five to ten times greater than the baryonic matter. One way to create such an equilibrium is to use the inverse-square law and Hooke's law. Below we use these laws to limit how deep the spacetime dent gets. We treat it like an elastic surface that springs back into place when mass is removed.
At equation 21 above, the two main variables are M (total galactic mass) and x (how deep is the dent). As dark matter is added, M and x grow proportionately (see 19), but as x adds to the dent's magnitude (equation 21's second term), it has an inverse-square effect (equation 21's first term) and the gravity is reduced until equation 21 equals zero--equilibrium is achieved (see 23 below). Using a pinch of algebra, we calculate the final mass of the galaxy (see equation 28).
We see the total mass is the dark-matter constant (we set to 10) plus 1 times the original mass (Mo) we used to break the symmetry. If we assume that baryonic matter is a catalyst for the build up of dark matter within a galaxy, then we can predict that galaxies with few stars, i.e., a small amount of baryonic matter will have little or no dark matter. And we can predict that galaxies that are nearly exclusively dark matter, have or have something equivalent to a massive black hole at their center. Finally, we predict that normal galaxies with lots of dark matter also have far more stars than galaxies lacking dark matter. These predictions lead to the final equation:
Where the radius (r) is small, equation 30 reduces to Newton's equation, but when r is large, dark matter rules!