# Does a planet/star itself lose mass when it 'emits' gravitational waves?

Orbiting planets and stars can create gravitational waves, as seen with the LIGO. But are they also losing mass-energy. Beside that, the kinetic energy associated with their orbital angular momentum is reduced.

For instance, in September 2015 the LIGO detected the merger of two black holes with masses of $35^{+5}_{-3}$ times and $30^{+3}_{-4}$ times the mass of the Sun (in the source frame), resulting in a post-merger black hole of $62^{+4}_{-3}$ solar masses. The mass–energy of the missing $3.0±0.5$ solar masses was radiated away in the form of gravitational waves.

So it looks like 3 solar masses were lost. But is that mass coming from the stars itself, or from their kinetic energy?

You've forgotten an important player in the system: the gravitational field.

Here's a pretty argument that gravitational fields are physically meaningful objects that carry energy: imagine two masses accelerating towards each other from rest, from a great distance away. The rest energy of the system is $E_\text{rest} = (m_1+m_2)c^2$; the kinetic energy is $K\approx\frac12m_1v_1^2 + \frac12m_2v_2^2$, at least while things are nonrelativistic, and only increases as a function of time. We introduce an internal energy $U=-Gm_1m_2/r$ so that we can make statements like "the total energy of the system is constant in time."

Now let's make partitions of our system to see whether we can account for everything. Looking only at the first particle, we see a total energy $E_\text{1} \approx m_1c^2 + \frac12m_1v_1^2$ which starts off positive and grows larger in time. Looking only at our second particle we also see a total energy which starts off positive and grows larger in time. So apparently if we only consider the particles in our system, we can't duplicate our statement that the total energy of the system is a constant in time. We need also to account for the energy tied up in the interaction between the two particles: the gravitational field. In electrodynamics and in general relativity you learn to actually compute how much of this interaction energy $U=-Gm_1m_2/r$ is found in any particular volume of the space around your interacting objects.

When objects emit gravitational radiation without colliding, that radiated energy comes from the gravitational field. Perhaps better, gravitational radiation is a redistribution of the energy stored in the gravitational field: energy is removed from the field near the interacting particles, leaving them more tightly bound to one another, and appears at large distances from them, where it can do things like move interferometer mirrors.

When you have nonrelativistic objects collide, you have conversion of gravitational energy into other forms of internal energy, like heat; this is why asteroid impacts can melt things. Eventually the heat gets radiated away, too.

A black hole is an object whose total energy is stored in the gravitational field --- we talk about a black hole's mass as a shorthand for how much of this gravitational energy there is.

• I’ve come across your interesting answer but I’m still confused. I’ve read many answers on this site which say that energy is not conserved in GR and Gravitational Potential Energy doesn’t exist in GR. So, when two masses accelerates towards each other due to gravity, does Gravitational energy stored in the field decrease according to GR? (By the word decrease, I mean less Gravitational energy that can do work or can be converted into other forms of energy) Thanks. – parker Feb 3 '18 at 17:01
• I would say that the emitted energy comes from the gravitational field, rather than from the rest energies of the massive objects involved in the orbit. This makes sense in the weak-field limit where it's still okay to talk about gravitational potential energy as a useful approximation. This might make an interesting follow-up question. – rob Feb 9 '18 at 18:24

Let me try to explain this by making an analogy with a simpler system i.e. a hydrogen atom. If you measure the mass of a hydrogen atom you find it is less than the mass of an electron plus the mass of a proton. In fact it is 13.6eV less.

This happens because if you let a separated electron and proton fall together under their mutual electrostatic attraction then when they meet they'll be travelling at a very high speed. In fact their speed will be too high for them to bind together. What happens is that they shed their excess energy by emitting a photon of energy 13.6eV, and that reduces the mass of the bound system by the equivalent mass given by Einstein's famous equation $E=mc^2$.

The hydrogen atom is a simple system, but this is quite generally true. The mass of a bound system is always less than the mass of its components if you separate them to infinity.

Now, a binary black hole is rather different because it doesn't necessarily have to shed any energy to form a bound system. If you collide two black holes exactly head on then they can merge straight into a single black hole and all the energy from their collision is trapped behind the new event horizon and can't get out.

However if the two black holes spiral towards each other then they will behave like a hydrogen atom. In order to form a bound system with a smaller and smaller radius they have to shed energy. That energy is shed as gravitational waves, and if you divide the energy they carry away by $c^2$ you get the amount the mass has decreased.From memory it's possible for up to a third of the total mass could be lost, though I wouldn't swear to that.

The natural question is to ask exactly how that mass is being lost, but that doesn't have a simple answer. It isn't matter turning into energy i.e. if you count the number of electrons, protons and neutrons going in then that number stays constant (just as it does when a hydrogen atom forms).

You could argue that it is, as you suggest, coming from the kinetic energy of the two black holes i.e. as they emit gravitational waves they slow down but it isn't really that simple. Basically the mass of a bound system is not simply the sum of the masses of its constituents, and it cannot be simply separated into contributions from the separate constituents.

• Do you think that the masses of the two collided BH mentioned in my question is including their kinetic energy? – Marijn Jul 7 '16 at 15:01
• You say: " It isn't matter turning into energy i.e. if you count the number of electrons, protons and neutrons going in then that number stays constant". But imagine that gravitons exists is it possible that they derive from matter of the BH like photons from a hydrogen atom? – Marijn Jul 7 '16 at 15:06
• @Marijn: the coupling of gravity to matter is exceedingly weak. It's possibly that in a unified theory linking the Standard Model and gravity there would be processes that convert matter into gravitons. However they would be so improbable that they are utterly negligible in a binary black hole merger. – John Rennie Jul 7 '16 at 15:26
• The masses of the black holes reported by LIGO are the masses when the black holes are widely separated, where widely means separated enough that gravitational wave emission hasn't made any significant change to the total mass of the system. – John Rennie Jul 7 '16 at 15:27
• I asked for the components of the BH as if in their mass was no kinetic energy, due to their speed, included than the negative result of three sunmasses should be the result of only the proper masses of the BH's. To make the sum: 35+30=65-62=3 – Marijn Jul 7 '16 at 15:49

As the question already beyond exact science, I will take a stab at it.

There are three components here - Kinetic Energy (KE), Energy of gravitational waves (GW), and lost mass in merger.

KE and GW -

As the GW are ripples in space (time), they have to be generated by motion and/or its disruption (i.e stoppage at moment of merger). Therefore, the energy of GW has to come out of the KE lost. Thus KE lost should be >= GW energy. As the BH are moving at speeds comparable to that of light just before merger, there should be plenty of KE to generate GW. I can not grasp creation of any ripples without use of some kind of motion?

Lost mass -

This is the one that is hard to explain. How mass is measured? For black holes, mass would be measured due to their gravitational effect. The lost mass means reduced gravitational effect of the merged matter. Therefore question comes back to gravity. May be that combined matter does not have same gravitational effect as sum of that of its separated components. The -ve potential energy acts as equivalent -ve mass making the net sum less by that amount (per E = M * C * C). This -ve mass would also show in gravitational effects which we use to estimate mass of the merged.

In other words, -ve potential energy = gained Kinetic Energy.

1. Significant amount of this KE is lost in space, creating ripples (GW).

2. Some should be absorbed by merged system in the form of rotation, heat etc.

3. Some may be lost as radiation.

Lost KE (= GW + radiation) is the equivalent lost/reduced mass.

Reduced mass via GW does not mean reduced matter, it means reduced gravitational effect.

In case of black hole merger, the rotational speed is slowed down by space itself, otherwise, their approach speed would reach escape velocity, which would be c. It seems, at highers rotational speeds, space starts absorbing the speed making it impossible to reach c. This phenomena probably gives the impression of infinite mass as the speed approaches c. Because any speed you impart, is absorbed by space and speed does not increase.

This absorption of rotational speed by space is the mechanism responsible for creation of GW. This must be the mechanism also for loosing mass via GW.

Linear speed would not be absorbed by space and, so there should not be GW if black holes collide head on, except that some GW may be possible due to sudden stoppage of the motion at collision moment.

Above is the case in events like merging of black holes.

In smaller events, like esteroid hitting a planet, there may be extremely little ripples/radiations and all (most) KE is absorbed by the merged system as heat/rotation and so, the lost gravitational effect may not be measurable.