# How does adding energy to an object affect its mass?

I probably learned that back in university when studying general relativity, but I forgot so I want to ask: As far as I understand it, any energy concentration creates a gravitational field, whether that energy is matter, electromagnetism or other energy, right?

The unit of gravity is meters per second^2.

Issac Newton's Law of Universal Gravitation tells us that the force of attraction between two objects is proportional the product of their masses divided by the square of the distance between their centers of mass - but this only accounts for mass and no other energies.

So what I want to ask is: if I add energy to a body not by adding matter but, for example, by heating it up, then based on my understanding the gravitational pull it creates increases too.

But does that also mean that its mass has now increased? Is mass solely defined via the gravitational field generated at a certain distance? Are they equivalent?

• Acceleration - gravity- is measured in meters per second per second. Sep 4 '20 at 18:18
• of course - typo Sep 4 '20 at 23:25

## 2 Answers

Spacetime is curved by energy and by mass (since mass contains energy). This means that a object that is heated up will be more massive and curve spacetime more than it did when cold. We will experience that curvature increase as an increase in its gravitational pull.

Newton's law of gravity doesn't deal with energy as a source of gravity; that discovery had to wait for Einstein, so it's not relevant in this context.

• How does this work with a planet that is tidally locked towards it star? Eventually one side of the planet would become very hot, and the other side very cold. Would the hot side of the planet have greater gravity than the cold side? Would this shift the centre of gravity towards the hot side? And further to this, when they were calculating the mass of our planets by using their orbits, did they take into account that Mercury and Venus are much closer to the sun, and warmer, than Saturn and Uranus? Sep 4 '20 at 19:53
• heat energy has a teeny-tiny effect on mass, so small that the issues you brought up wouldn't be anywhere close to being measurable. Sep 4 '20 at 20:17
• Thanks. c is so large that under e=mc^2 it would take a huge energy to make any measurable change compared to a change in mass. And by the way, teeny-tiny is exactly the wording that I can understand. Sep 4 '20 at 20:37
• but the question remains - does adding energy (in the form of heat, kinetic energy, electrical charge) change the mass of the object or is 'mass' solely a property of the amount of matter present? Sep 4 '20 at 23:28
• adding energy changes the mass. Sep 5 '20 at 3:16

The main equation of GTR is $$G=8\pi T$$ where T is the stress-energy tensor. T contains contributions from energy/momentum of matter and fields.

Yes, $$E=mc^2$$ heating up an object or accelerating it increases it's mass, minutely.

Is mass solely defined via the gravitational field generated at a certain distance? Are they equivalent? It is quite difficult to define the exact nature of mass and energy - for example, Noether's theorem suggests energy is a quantity conserved as a result of invariance of equations under time displacement. Hence one would expect the choice of "zero" energy to be arbitrary, but $$m$$ appears as a absolute quantity in Gravity. Energy isn't necessarily conserved in GTR either. Probably best to go with the standard kilogram definition of mass. :-)