# Inertial vs gravitational mass at different temperatures

So, I know more energetic objects have more mass according to Einstein. I'm aware that this has been verified by weighing an object at different temperatures where the object weighed more at higher temperature.

I'm also aware that gravitational mass and inertial mass are the same, as verified by a number of experiments, IE dropping a bowling ball and feather, which fall at the same rate.

Thing is, I am not aware of a drop comparison between two objects at different temperatures. The thought that a difference could emerge goes against our known physics, but I'm just wondering if this has ever actually been tested.

I've deleted my original answer, because it didn't really address your question and I provided the wrong information at first.

In a comment on the deleted answer, you said "I'm concerned with the rate of acceleration on the way down." This is true in the (outdated) Newtonian conception of gravity, but in general relativity it isn't falling objects that accelerate, it's ones that do not fall. This seems counterintuitive at first, but is more in accord with experience and experiment. When a car accelerates forward, you can actually feel the acceleration as pressure on your back, and an accelerometer will display the amount of acceleration. Similarly, when standing on the ground you can feel the Earth pushing against your feet. An accelerometer placed on the ground will register 9.8 $$m/s^2$$ acceleration.

Conversely, a dropped accelerometer shows no acceleration at all, and while you are falling you feel no forces on you -- there's no way for you to tell whether you are falling near a planet or floating in deep space, except by looking.

In general relativity this is all explained as objects following geodesics (the analog of "straight lines") in curved spacetime. Only if a force acts on an object and causes it to accelerate does it deviate from a geodesic. Free falling objects are not acted on by any forces (gravity is not a force in relativity!) and follow the paths dictated by spacetime, which is curved by the Earth. Objects on the ground are acted on by forces (the contact forces between themselves and the ground) and that is why they no longer fall.

General relativity thus explains why objects of different masses fall in the same way -- they are just following the natural path dictated by (curved) spacetime. In order for a hot object to follow a different path than a cold object, there would have to be some additional force acting on the hotter object. There is no evidence for any such forces, and a variety of reasons to think there are not any. GR has been tested in a variety of high energy / high temperature situations (black hole collisions, planets orbiting near stars, etc.) and has held up to all tests so far.

Objects themselves bend spacetime ("matter tells spacetime how to curve, and spacetime tells matter how to move") and the particular configuration of an object's energy, including its pressure and temperature, does affect how spacetime is curved near it. So your question is a good one. But there is a mathematical theorem that says that vacuum solutions of the Einstein field equations (i.e. the way that spacetime is curved outside of the matter curving it) are limited to certain special cases, and pretty much only depend on the energy and angular momentum of the matter in question. So the matter's temperature affects distant objects only in the way any other energy would.

• So, I get GR and space-time geodesics, I think. Which is why I said the objects falling at different rates based on their temperatures would go against our current understanding of physics. It is probably clear from my question that I'm a little bit skeptical of GR. I get GR says objects don't experience a force when they are falling. But acceleration can only be define with respect to another reference frame. It doesn't in itself imply a force, nor can it be expressed in absolute terms. So I think it is accurate to say objects accelerate toward earth when falling according to GR. Dec 8, 2021 at 23:49
• No, the acceleration I'm talking about ("proper acceleration") is not relative. It's something that can be felt, and can be measured by an instrument (an accelerometer), so it does not need to be defined with respect to a coordinate frame. You can define a "coordinate" acceleration that's relative to a frame, but that's purely a human convention with no physical meaning, whereas proper acceleration is physical and objective. Dec 9, 2021 at 0:39
• Also, I can't let the "little bit skeptical of GR" comment slide :). GR has over a century of experimental evidence backing it up. It's been worked on by some of the greatest mathematicians and physicists in history, including people like Hilbert, Noether, Goedel, Schroedinger, Hawking, and Feynman. Someday an even better theory of gravity may come along, but right now GR is the best we have, and it is unquestionably far better than Newton's theory of gravity. Dec 9, 2021 at 1:58
• I take your point regarding proper acceleration. But unfortunately, I don't think this really gets us anywhere. You can't question the mechanization of GR from within the confines of GR. Dec 9, 2021 at 4:16
• So, yeah, I know that GR is objectively, the most accurate model of gravity we have. But we do know the story of gravity is incomplete and/or wrong in some way. Maybe dark matter exists, and maybe quantum gravity can be sorted out, all without rewriting the 'how' of GR. We once thought the planet Vulcan existed, only to learn our understanding of gravity was wrong. And at the time, Newtonian gravity was an incredibly successful theory, and it was developed by what was probably one of the greatest minds in history. Dec 9, 2021 at 4:21