Dark Matter vs. Mass from Kinetic Energy

Question

I was thinking about dark matter, and was wondering if the extra mass due to kinetic energy has been taken into account.

Here's what I mean: let's talk about any cosmological object that is known to have dark matter. Now, that object is moving away from us at a certain velocity, plus, the object might be moving on an axis within itself. So it has a certain kinetic energy ($E=\frac{1}{2}mv^2$) as seen from our perspective.

The mass-energy equivalence ($E=mc^2$) tells us how much more apparent mass is added due to this energy. Therefore:

$$ m_{extra}=\frac{E}{c^2} $$

$$ m_{extra}=\frac{mv^2}{2c^2} $$

I am aware that the formula for kinetic energy that I am using here is an approximation, and is not accurate that the scales that I am talking about. I am simply using it to demonstrate my thought process.

My question is, is this "extra" mass usually taken into account when calculating the mass for objects where this might be significant? Can this possibly explain some of the observed dark matter?

The formulation above can clearly not account for ~5x the observed mass, unless $v$ is about $3.16c$. It's an approximation, and we'll probably need to take into account the energy from various kinds of motion to get a more accurate picture.

It seems to me that this should be significant because all of the objects/phenomena that are cited as evidence for dark matter have one or both of these properties:

1. They are usually far away from us (no dark matter has been observed on Earth, or in our neighborhood, at least not significant amounts that I am aware of.) Seems obvious from the formulation above, since the relative velocities are not large enough.
2. They sometimes have extremely high internal velocities (for example, the Bullet Cluster.)

Answer 1

You need to be careful when talking about the extra gravitation created by kinetic energy. It's tempting to think that if you look at a single object then because its relativitic mass is given by:

$$ m_r = \frac{m}{\sqrt{1 - v^2/c^2}} $$

if you make the object move fast enough it will generate a bigger and bigger gravitational field and even eventually turn into a black hole. However this does not happen. The reasons are discussed in several questions, e.g. If a 1kg mass was accelerated close to the speed of light would it turn into a black hole?, so here all I will add is that this is a good example why relativistic mass is a misleading concept that is no longer used outside of the less informed popular science TV programmes.

However kinetic energy will increase the strength of a gravitational field in some circumstances. Suppose you take a ball of cold gas the mass of the Sun and you add energy $E$ to it to heat it up. This increases its mass by an amount $\Delta m = E/c^2$ and does increase the gravitational field. The energy $E$ goes into increasing the kinetic energy of the gas particles in the ball, so in this case increasing kinetic energy does increase the gravity.

It isn't clear from your question which of the two cases you are suggesting - possibly both. A galaxy cluster doesn't get heavier just because it's moving relative to us, so that can't explain dark matter. However if the gas/stars/whatever inside the cluster have appreciable kinetic energies that does add to the mass and that needs to be taken into account.

However it should be obvious that kinetic energy of stuff inside a cluster cannot account for the extra mass or indeed get even close to it. Remember that when we talk about kinetic energy of stuff inside a gravitationally bound object we are in effect talking about its temperature. To get the internal energies high you need to heat the cluster, and long before the extra energy became significant the cluster would simply evaporate.

For a gravitationally bound system there is a nice simple relationship between the kinetic energy $T$ and potential energy $V$ in called the virial theorem:

$$ T = -\tfrac{1}{2}V $$

and this sets a limit on the kinetic energy that is many, many orders of magnitude too small to account for the observed dark matter density.

Answer 2

The clue is in the name - cold dark matter. A working definition of cold here, is that the total energy is roughly equal to the rest mass energy. Or in other words, that the kinetic energy is very small compared with the rest mass energy.