energy - Can change in temperature cause a change in mass of an object?

If a gold bar is heated to say 200 degree Celsius then will it have the same mass at say 10 degree Celsius. Does energy has mass? If so then does this increased 'heat energy' cause an increase in the mass of an object

Answer

A gold brick is made of gold atoms that mutually interact. At a given temperature it weighs bit less than the weight of each piece. At a higher temperature it weighs a bit more than at a colder temperature. So the weight isn't the sum of the weights of the parts, not quite.

Even in a single gold atom, it weighs a little bit less than weight of each neutron and proton and electron all added up. And even for a lone proton or neutron the weight isn't the sum of the weights of the quarks inside.

Why? Well, firstly weight isn't caused by mass, it is caused by energy and momentum and pressure and stress. But there is energy associated with mass and for many systems that source, the energy associated with mass is the largest, so large by far that you get almost the correct answer by adding up just that.

But there are other meanings of mass. Does a big chunck of gold resist forces more when it is hotter? Yes it does.

The way you resist forces is related to how you balance energy and momentum. There is energy associated with mass and there is energy associated with momentum. For a single particle $E^2=c^2\vec p^2 + m^2c^4$. So at first when you add a bit of momentum the energy increases by about $ \vec p^2 /2m$ but later after you've added lots of energy from momentum a little bit of additional momentum $\Delta \vec p$ adds about $\vert c\Delta \vec p \vert$ of additional energy. OK, so for a single particle the mass is about how you balance energy and momentum.

Now, if you have a system with a total system energy of E and a total system momentum of $ \vec p$ then the system can act like a giant particle of mass $\sqrt{E^2-c^2\vec p^2}/c^2$. And now you can see that heating the gold brick in the frame where it has no total momentum doesn't increase the total momentum (heating gives more momentum equally in all directions so the total momentum stays zero) so it increases the total system energy and so the system starts to act like a giant particle with a larger mass.

Since that balance of energy and momentum is what determines the kinematic and dynamical aspects of mass like how hard you have to push to get it to go faster the hot brick has a bigger mass.

So by every possible sense. It has more mass. It's still a composite object so the mass of the system (the brick) is not the sum of the masses of the parts. But we've learned that that never happens anyway so we can deal.

So wait you might say. If the mass or weight is never actually the perfect sum of the masses or weights. At least why does it appear to be the sum?

Good question. Imagine that for each particle you had a vector in spacetime with components $(E/c, p_x, p_y, p_z)$ its like a 4d momentum. It turns out that it points in the 4d direction the particle travels in spacetime, and for two particles moving the same way in space time (so traveling from point 1 at time 1 to place 2 at time 2) if one has twice the mass it has twice the 4d momentum. So in a sense the mass is the length of the 4d momentum and the direction is the direction in spacetime the particle is going.

But if you have a bunch of particles that are all moving slow ... relative to each other. Then in a geometric amount of time they all moved almost the same way. So in fixed time interval they all ended up at really close places, so those directions in spacetime are all pointing in almost the same direction.

So their masses are the lengths of their 4d momentums and the 4d momentums are pointing in almost the same directions.

Here is an insight from geometry. The sum of vectors has a length that is almost exactly the sum of the lengths when they are all pointing in almost the same direction. If they are moving slowly relative to each other then those 4d vectors are pointing in almost the same direction so he length of the sum is close to the sum of the lengths. And if you sum those vectors you get the total energy and the total momentum.

So the mass is really close to the sum of the masses when everything is moving slow relative to each other. And in our daily experience that is what happens even a performance race car or high speed jet is slow compared to light. And when we compared the size of that fixed time interval to the different places of the particles we compared the ratio to the speed of light so for everyday life it was a small difference.

There is one difference between usual length and the mass as length of a vector idea I showed. The mass is $\sqrt{E^2-c^2\vec p^2}/c^2$ which has a minus sign. But it is like a length and the geometric fact that the sum of vectors has a length that is almost exactly the sum of the lengths when they are all pointing in almost the same direction, it still holds when you use this kind of a way to measure lengths.

So, yes a hot brick weighs more because of the increased energy. And it resists motion more because you increased the energy by distributing the additional momentum so that in the frame where the total momentum was zero it continues having a total momentum of zero, so the energy in that frame could go to increasing the mass of the system (the length of the 4d momentum of the system).