# Does E=mc^2 apply to gravitational potential energy?

## Summary:

does the mass of a falling object increase, decrease, or stay the same?
I'm reading Schutz's A First Course In General Relativity and in chapter 5 he discusses an idealized experiment in which an object is dropped from a tower, then turned into a photon and sent back up to its original height.

In classical mechanics we would say that as the object falls it loses potential energy and gains equal kinetic energy so at every moment its total energy is constant. In relativity, if an object gains energy its mass increases according to $\Delta m = \Delta E / c^2$. Does this apply to both kinetic and potential energy so that — as in classical mechanics — the object's mass remains constant as it falls?

On the one hand, energy can't just disappear and as I understand relativistic mechanics, $E=mc^2$ applies to potential energy as well as kinetic. On the other hand, the whole point of the tower thought experiment is that the object has more mass[energy] after it has fallen to its lower position.

Dale
Mentor
2020 Award
Relativistic mass is a concept that has been largely discarded. When physicists today just say "mass" they mean the invariant mass, not relativistic mass.

Abhishek11235, Demystifier and vanhees71
Ibix
2020 Award
Gravitational potential energy (to the extent it can be defined in relativity) isn't a property of the body, but of its interaction with the gravitational field. It's very difficult to pin down "where" gravitational energy is, but you can loosely think of it as a property of the field rather than the body.

So the total energy of the body is higher when it's moving relative to a hovering observer than when it isn't. Probably best not to think of that as mass, though, since GR is formulated in terms of invariants.

mfb and etotheipi
Staff Emeritus
I think what people are saying is that your question is a hodgepodge of Newtonian mechanics, SR and GR. It's probably unanswerable in its present form. You might think of rephrasing.

By the way, if you are 5 chapters into Schutz and are still thinking in terms of relativistic mass and energy as "localizable stuff", you are likely to have an uphill slog.

waitshift and vanhees71
Nugatory
Mentor
In relativity, if an object gains energy its mass increases according to Δm=ΔE/c2.
You'll hear that a lot, but it is not quite right, and the inaccuracy can lead to all sorts of misconceptions. It is more accurate to say that if a system with zero total momentum gains energy its mass (necessarily rest mass, because we're talking about a system with zero momentum) will increase according to ##\Delta{m}=\Delta{E}/c^2##. Because ##c^2## is the conversion factor between measuring energy in units of kilograms and in units of Joules (or whatever other units you choose), this statement is basically a tautology: If a system gains energy it gains energy.
Does this apply to both kinetic and potential energy so that — as in classical mechanics — the object's mass remains constant as it falls?
Here the system of interest is not the object, but the object and the earth. If we were to enclose the entire earth/object system in a single huge box, we would find that the mass of the box and everything in it remained constant: the kinetic energy of the object goes up as it falls, the potential goes down by the same amount, the total mass/energy inside the box is unchanged.

On the other hand, if we were to reach into the box and apply a force to lift the object away from the earth (strictly speaking we would have to apply an equal and opposite force to the earth to maintain the condition of zero total momentum) then we're adding energy to the system from outside by doing work on it, and we would find that the mass of the box will increase accordingly.

But note that there is no way of associating this mass increase with the earth or the object and saying that the mass of of either has increased. It's a property of the system as a whole, the box and its contents.

Last edited:
Karl Coryat, Bandersnatch, vanhees71 and 3 others
Thanks all, I appreciate your efforts.

Let me bring this down to Earth: Suppose I have a labratory which includes a tower, and I drop an object from the top of the tower, and — using measuring devices that are at rest relative to the tower — I measure the object's mass as it falls. It sounds like, based on

So the total energy of the body is higher when it's moving relative to a hovering observer than when it isn't.
that I would in fact measure an increase? (It's fine with me if we call this energy instead of mass.)

Ibix
2020 Award
Suppose I have a labratory which includes a tower, and I drop an object from the top of the tower, and — using measuring devices that are at rest relative to the tower — I measure the object's mass as it falls. It sounds like, based on

that I would in fact measure an increase?
No, because "mass" means invariant mass, which doesn't change by definition. If you measure the relativistic mass, then yes that will increase by definition.

Dale
Mentor
2020 Award
It's fine with me if we call this energy instead of mass.

vanhees71 and etotheipi
PeterDonis
Mentor
2020 Award
I would in fact measure an increase?
What are you measuring, and how are you measuring it? Saying...

using measuring devices that are at rest relative to the tower — I measure the object's mass as it falls
...is pretty vague. How are you going to measure the mass of something that's moving relative to the measuring device?

Staff Emeritus
I measure the object's mass as it falls.
How do you do this? Surely not by grabbing it and putting it on a scale.

vanhees71
pervect
Staff Emeritus
Thanks all, I appreciate your efforts.

Let me bring this down to Earth: Suppose I have a labratory which includes a tower, and I drop an object from the top of the tower, and — using measuring devices that are at rest relative to the tower — I measure the object's mass as it falls. It sounds like ... that I would in fact measure an increase? (It's fine with me if we call this energy instead of mass.)
The energy of a moving object is higher than the energy of an identical stationary object. Since you specified "measuring" it, you might want to sketch out how you propose to do your measurment. Alterntively, you can treat it as a question about defintions, rather than experiment, and say that by definition, if you have two identical objects, one moving, and one stationary, the moving object has more energy.

Note that this implies that the energy of an object is not solely the property of the object - you can't assign an energy to an object until you know whether it's moving or not, which means you need to specify the frame of reference of an object before you can know it's energy.

etotheipi and snoopies622
Nugatory
Mentor
which means you need to specify the frame of reference of an object
Somewhere between a quibble and scratching a pet peeve.... but should that not be “the frame of reference you have chosen to use”?

etotheipi
This reminds me of a question I asked here years ago: If I put a mousetrap on a scale and then set the mousetrap, would it weigh more? The answer came back: In principle, yes.

vanhees71
PeterDonis
Mentor
2020 Award
If I put a mousetrap on a scale and then set the mousetrap, would it weigh more? The answer came back: In principle, yes.
Yes, because you setting the mousetrap requires reaching in from the outside and doing work, which adds energy to the mousetrap, for the reasons @Nugatory gave in post #5.

Consider an alternative scenario: you have a mousetrap with a charged battery and a battery-operated mechanism that sets the mousetrap, controlled by a timer. Before the timer triggers the mechanism, you put the mousetrap, including the battery-timer-mechanism, on a scale. When the timer triggers and the mechanism sets the mousetrap, the weight recorded by the scale does not change, because the energy used to set the mousetrap is now coming from something internal to the system--the battery--instead of being put in from outside.

vanhees71 and etotheipi
Let me bring this down to Earth: Suppose I have a labratory which includes a tower, and I drop an object from the top of the tower, and — using measuring devices that are at rest relative to the tower — I measure the object's mass as it falls.

Let's say that at the top of a tower a pea is turned into light, which is sent down to the middle of the tower, where the light is turned into a potato, which is then turned into light, which is sent down to the ground floor of the tower, where the light is turned into a melon.

Let's say that the locally measured inertial mass of the pea is one gram, the locally measured inertial mass of the potato is 100 grams, and the locally measured inertial mass of the melon is 10 kg.

(A more complete description of the thought experiment in the book would be nice)

Last edited:
PeroK
Homework Helper
Gold Member
2020 Award
Let's say that at the top of a tower a pea is turned into light, which is sent down to the middle of the tower, where the light is turned into a potato, which is then turned into light, which is sent down to the ground floor of the tower, where the light is turned into a melon.

Let's say that the locally measured inertial mass of the pea is one gram, the locally measured inertial mass of the potato is 100 grams, and the locally measured inertial mass of the melon is 10 kg.
Is that from a first course in physics by Monty Python?

vanhees71, etotheipi and Motore
PeroK
Homework Helper
Gold Member
2020 Award
Thanks all, I appreciate your efforts.

Let me bring this down to Earth: Suppose I have a labratory which includes a tower, and I drop an object from the top of the tower, and — using measuring devices that are at rest relative to the tower — I measure the object's mass as it falls. It sounds like, based on

that I would in fact measure an increase? (It's fine with me if we call this energy instead of mass.)
Here's the real issue. If you were ready for the Schutz book, you would already know that some older texts use the concept of relativistic mass while modern texts generally do not. You would have a reasonable grasp of why this is and - although ultimately it's a matter of taste - the advantages of avoiding relativistic mass.

You would also be able to laugh at the pop-science obsession with relativistic mass as a cornerstone of SR.

That you are unsure about this suggests that you need significantly more study of SR before you tackle GR.

etotheipi
vanhees71
Gold Member
That's really interesting. I thought that the very inconvenient theory of "relativistic mass" is only used in some old-fashioned textbooks on SR. You can get away with this confusing concept in SR and that's why unfortunately it's still used even in some newer textbooks.

It's new to me that this concept is also used in GR textbooks. I cannot conceive how you get away with this, using non-covariant concepts in GR. Does Schutz really use "relativistic mass" in GR? I cannot believe it, before I've seen it!

Ibix
2020 Award
Does Schutz really use "relativistic mass" in GR?
No, at least not on page 112 which discusses this "drop a massive particle, convert it to light and shine it back up" way of deducing the existence of gravitational redshift. He uses "rest mass" and "total energy". Nor does the term appear in the index.

vanhees71, Dale and PeroK
Staff Emeritus
Does Schutz really use "relativistic mass" in GR?
He does not. He does say "rest mass" in one place.

vanhees71
vanhees71
Gold Member
I'd have been very surprised by the opposite. I've no clue how to use a concept like relativistic mass in GR, as it seems very difficult, if not dangerous, to work with non-covariant objects. One must not forget that the general covariance is a gauge symmetry, and to introduce objects with unclear definitions of how to transform under diffeomorphisms of the coordinates is thus very dangerous.

pervect
Staff Emeritus
I'd have been very surprised by the opposite. I've no clue how to use a concept like relativistic mass in GR, as it seems very difficult, if not dangerous, to work with non-covariant objects. One must not forget that the general covariance is a gauge symmetry, and to introduce objects with unclear definitions of how to transform under diffeomorphisms of the coordinates is thus very dangerous.
I don't have Rindler's textbook, but I've read it in the past, in response to some other posts by some long-gone (banned from PF for incorrect claims) relativistic mass enthusiasts.

From memory, I believe Rindler was fond of talking about mass as the ratio of 3-acceleration to 3-force. Rindler used his concept of mass to motivate the stress-energy tensor, and he'd describe the effect as a rod under pressure as having more "mass" (by his defintions) depending on the longitudinal pressure.

Understanding the 4-dimensional stress-energy tensor - or at least accepting it well enough to work with it - is a key requirement to proceed in General Relativity, one that I believe many students struggle with. Textbooks, for the most part, don't really try to motivate the stress-energy tensor, though they all present it and show how to use it.

I do remain convinced that it is both simpler and more helpful to students to introduce other 4-dimensonal objects, such as the 4-dimensional 4-velocity and the 4-force, before going on to the 4 dimensional stress-energy tensor. But of course students and readers have their own ideas. I tend to believe that most students who refuse to learn about 4-vectors cripple themselves, but there is little I can do about it other than to suggest that they not do that.

The tie in to "relativistic mass" here is that the 4-vector approach naturally leads to the invariant mass, which is the "length" of the 4-vector, and that "relativistic mass" in the 4-vector formalism is just one of the compnents of the 4-vector, namely the energy component of the energy-momentum 4-vector.

Using relativistic mass is in my mind associated with not having learned or refusing to learn about 4-vectors. Presumably this seems easier to the student at the time, but I believe in the end it hurts their understanding. Anyway, to my mind, the whole relativistic mass issue is a speedbump on the way to understanding the stress-energy tensor. It's not so important in and of itself, because the whole idea of mass is going to wind up being replaced by other notions anyway. At least it well if things go well. Sometimes things do not go well.

Einstein's field equations do not describe gravity in terms of "mass", but in terms of the stress-energy tensor.

Anyway, there is a small possibility that someone who simply cannot let go of "relativisitc mass" might be helped by Rindler's old textbook. But in general I think that the best thing to do is to learn the 4-vector approach.

vanhees71 and Dale
In classical mechanics we would say that as the object falls it loses potential energy and gains equal kinetic energy

Let's first put a photon in a box at the top of a tower and then lower the box down to the bottom of the tower. Then let's shine the photon back up and lift the box back up.

A box that had gravitating energy ##E_{photon} + E_{box}## was lowered.

A box that had gravitating energy ## E_{box}## was lifted

Clearly the agent that lowered and lifted, gained some net energy in the process.

Now, as I want to stay in the mainstream, I must say that during the lowering the lowering agent took energy from the box-photon-planet system. And when the photon climbed back up the photon lost energy to the box-photon-planet system.

Now I have a problem: Should I say that when the photon climbed back up it gained potential energy, or should I say that it did not gain potential energy?? Oh yes, it is the box-photon-planet system that gains potential energy, not the photon.

So, does the problem go away, if objects do not lose or gain potential energy when they climb or fall?

PeterDonis
Mentor
2020 Award
A box that had gravitating energy ##E_{photon} + E_{box}## was lowered.

A box that had gravitating energy ## E_{box}## was lifted
What do you mean by "gravitating energy"?

By that I mean, show us the actual math that backs up your claims here.