# How to prove gravitational mass & inertial mass equivalence?

• I
Tio Barnabe

## Main Question or Discussion Point

There is a video on YouTube where Sean Carroll says for Newton it was just an accident that inertial mass equals gravitational mass, but with the general theory of relativity it became obvious that it has to be so. How does one see that?

My own attempt has been consisting of transforming between inertial frames in the following way

Suppose there is a inertial frame A where the equation of motion of a particle is given by
$$m_{_I} a = m_{_G} g + F(x)$$ where $m_{_I}, m_{_G}$ stands for inertial and gravitational mass, respectively, $g$ is the (constant) acceleration of gravity and $F(x)$ a non gravitational, constant force.

If there's another frame B related to A by $x' = x - at^2 / 2, t' = t$, then the equation of motion for the same particle reads (as can be calculated)
$$m_{_I} a = m'_{_G} g + F(x)$$ The inertial mass $m_{_I}$ remains the same as for A because it's invariant according to Newton's theory; the non gravitational force $F(x)$ and $g$ also remains unchanged because they are vector quantities.

We must conclude from the above equation that $m'_{_G} = m_{_G}$, that is, the gravitational mass is also invariant under this particular transformation.

Of course, this doesn't show what I'm trying to realize, i.e. how does GR states that $m_{_I} = m_{_G}$ in any intertial frame.

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vanhees71
Gold Member
2019 Award
In general relativity it's not only obvious but just built in by construction. However, according to GR it's not mass that is the source of the gravitational field but the energy-momentum-stress distribution of all kinds of matter (including radiation).

Tio Barnabe
You basically repeated what Carroll said without given the reason behind.

Staff Emeritus
2019 Award
Tio, I'm afraid that nobody is going to be able to explain the inner workings of GR at the B-level.

Tio Barnabe
If this is the problem, I may copy-past my opening post to a new, I level thread and some moderator may delete this one....

Ibix
GR is built on the assumption that you cannot detect "acceleration due to gravity". That's the equivalence principle. The only way it makes sense is if the gravitational mass and inertial mass are the same thing for point particles.

Tio Barnabe
Alright. I'm going to answer my own question.

If the two inertial frames are related by $x' = x - (1/2) gt^2$ and $t' = t$. If the dimensions are small enough, the gravitational field acceleration $g$ can be taken constant.

In the unprimed system the eq. of motion reads $$m_{_I} \frac{d^2 x}{dt^2} = m_{_G}g + \vec{F}$$ In the primed system the eq. of motion reads $$m_{_I} \frac{d^2 x'}{dt'^2} = \vec{F}$$ using the given relation between the two systems, IFF the inertial mass equals the gravitational mass. So, gravity was replaced in one frame by a non-gravitational interaction in the another frame. By the Equivalence Principle this is so if $\delta g \approx 0$. So the earlier assumed equivalence between the inertial and gravitational mass has to be valid in general.

PeterDonis
Mentor
2019 Award
How does one see that?
Because in GR gravity is not a force, so there is no separate concept of "gravitational mass" at all. What we think of as "gravity" in Newtonian terms, in GR just becomes the effect of spacetime geometry on the motion of objects. So "gravitational mass" in GR is now just "how objects respond to the spacetime geometry", which is the same thing as inertial mass.

vanhees71
Gold Member
2019 Award
Tio, I'm afraid that nobody is going to be able to explain the inner workings of GR at the B-level.
Well, here in Frankfurt we offer lectures on GR for BSc students with quite some success. In the last semester we've had quite a number of students who took successfully the course in their 3rd semester.

Mister T
Gold Member
There is a video on YouTube where Sean Carroll says for Newton it was just an accident that inertial mass equals gravitational mass, but with the general theory of relativity it became obvious that it has to be so. How does one see that?
I think what Carroll is saying is that in newtonian physics we explain that all objects have the same free fall acceleration regardless of their mass using an equivalence of gravitational and inertial mass, but in GR we have a different explanation and that explanation doesn't involve that coincidence.

You seem to be looking for an explanation of that equivalence in GR, but instead perhaps you should instead be looking at how GR explains that all objects have the same free fall acceleration regardless of their mass.

Staff Emeritus