# Does the mass of a falling body decrease?

Let's say a body with m=2kg falls from 100 meters. Obviously it's speed would be far lower than the speed of light so the change in mass (if it exists) would be very tiny. However, I know that if the speed increases, its mass would increase too. That's because its kinetic energy would become bigger. On the other hand, its potential energy would decrease in the same amount that KE has increased.

Does this suggest that either the mass is not going to change (due to the conservation of energy) or it would become slightly bigger, but never less?

I assume you are talking about 'relativistic mass', i.e $m(v)=\gamma(v) m_0$. No one really uses this notion any more because it is not quite useful to reason about this quantity as if it is a mass, so I am going to talk about the energy $E=\gamma m_0 c^2$ instead.

By $E^2= m_0^2 c^4 + |\vec{p}|^2 c^2$ we see that the energy increases as the speed increases so it should get bigger as it falls downwards.

• Of course, it doesn't if you factor in the potential energy as well--Schwarzschild geodesics have a conserved energy. – Jerry Schirmer Feb 23 '13 at 22:13
• Yes, naturally. I don't think taking into account GR effects would be helpful in the context of this question though. Granted, Newtonian gravity + special relativity doesn't even begin to make sense but it seems that this question would arise in any context where something accelerates. (i.e I am not addressing the case where the body accelerates due to gravity per se) – alexarvanitakis Feb 23 '13 at 23:01

If you were observing the falling body alone, then it would appear to gain mass as its kinetic energy increases. However, a corresponding amount of mass would be lost from other parts of the falling object/Earth system. So, observing the falling body alone, it appears to gain mass. Observing the system as a whole, it does not.

My notion (although someone more knowledgeable about GR may correct me) is that the mass is lost from the graviton field between the two bodies.

• Graviton field? Is it different from the gravitational field? In what aspect does it make the body loose energy/mass? – menislici Jan 24 '13 at 1:14
• They are the (theoretical) particle that would carry the gravitational field. The gravitational field decreasing in energy/mass happens the same way that a decreasing electric field energy/mass happens, with fewer photons. While the photons/gravitons have no rest-mass, they do have relativistic mass since they carry energy. – KDN Jan 24 '13 at 1:20