# Modified gravity, demystified

Today we will talk about modified gravity.
This is a sequel to my earlier video about dark matter.
In the earlier video I explained why physicists believe that our universe contains dark matter,
but that some observations are difficult to explain with it.
In this video I will explain how modified gravity explains the observations.
Let us have another look at the galaxies, and the disk galaxies more specifically.
In disk galaxies most of the mass is in the center of the galaxy.
This means if you want to calculate how a star moves far away from the center it is a
good approximation to only ask what is the gravitational pull that comes from the center bulge of the galaxy.
Einstein taught us that gravity is really due to the curvature of space-time, but in
many cases it is still quantitatively accurate to describe gravity as a force.
This is known as the “Newtonian limit” and it is a good approximation so long as the
gravitational pull is weak and objects move much slower than the speed of light.
It’s a bad approximation, for example, close by the horizon of a black hole.
But it’s a good approximation for the dynamics of galaxies that we are looking at here.
It is then not difficult to calculate the stable orbit of a star far away from the center
of a disk galaxy.
For the star to remain on its orbit, the gravitational pull must be balanced by the centrifugal force.
You can solve this equation for the velocity of the star, and that will give you the velocity
necessary for the star to remain on a stable orbit.
As you can see, the velocity drops inversely with the square root of the distance to the center.
If you draw this relation on a graph it looks like this.
But this is not what we observe.
What we observe instead is that the velocities continue to increase
with distance from the galactic center and then they become constant.
This is known as a flat rotation curve.
This is not only the case for our own galaxy, but it's the case for hundreds of galaxies
that have been observed.
The curves don’t always become perfectly constant, sometimes they have wiggles but
it’s abundantly clear that these observations cannot be explained by the normal gravitational
pull caused by the normal matter only.
Dark matter solves this problem by postulating that there is additional matter in the galaxies,
distributed in a spherical halo.
This has the effect of speeding up the stars because the gravitational pull is now stronger
due to the mass from the dark matter halo.
There is always a distribution of dark matter that will reproduce whatever velocity curve
you observe.
In contrast to this, Modified Newtonian Dynamics – or MOND for short – postulates that
gravity works differently.
In MOND the gravitational potential is the logarithm of the distance and not, as normally,
the inverse of the distance.
In MOND, the gravitational force is then the derivative of the potential, so that’s inversely
proportional to the distance, and not as in normal gravity, inversely proportional to
the square of the distance.
If you plug this modified gravitational force into the force-balance equation as before,
you will see that the dependence on the distance cancels out and the velocities just become constant.
Now of course you cannot just go and throw out the usual one-over-R-squared force law
because we know it works on our planet and it works in the solar system.
So MOND postulates that the normal one-over-R-squared law crosses over
into a one-over-R law.
This cross-over happens not at a certain distance,
but it happens at a certain acceleration.
The new force law comes into play at low acceleration.
This acceleration where the cross-over happens is a free parameter of MOND, usually denoted
a with an index zero, or a-naught.
You can determine the value of this parameter by just trying out which value fits the data best.
It turns out that the best-fit value is closely related to the cosmological constant.
Why that is so, no one has any idea.
The cosmological constant is a specific type of dark energy, and there is no known relation
between dark matter and dark energy.
You can also use the above relation to find out what is the relation between the velocities of far-out stars
and the mass of the galaxy.
It turns out that the mass of the galaxy scales with the forth power of the velocity.
This is known as the Tully-Fisher relation and it is very well confirmed by observations.
Modified gravity predicts it.
Dark matter cannot predict it.
If you allow yourself to distribute dark matter in galaxies however you want, then you can
of course also get the Tully-Fisher relation by choosing whatever distribution it is that works.
If you however try to predict the distribution of dark matter by using computer
simulations for structure formation, then you will find that it is difficult to recover
what we actually observe.
If you want to recover what we observe with dark matter you will have to introduce a lot
of new parameters.
So you can make it work but not in any simple way.
MOND however explains the observations in a very simple way.
However, we already know that MOND is wrong.
MOND does not work well for the galaxy clusters and it does not work for
the early universe.
This should not surprise you because, as I told you the last time, MOND is only an approximation
that works in some cases, just like the Newtonian limit is only an approximation that works
in some cases.
Of course you would rather want to have a full theory to replace Einstein’s theory of general
relativity from which you can then derive MOND in a certain approximation.
This theory is referred to as modified gravity and we presently don't know
what this theory looks like.
\we have a few approaches but personally I don’t find any of them too convincing.
But next time I want to tell you what I think is presently the most convincing explanation
for out observations.