- #1

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.

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.