Gravitational mass

There are several distinct phenomena which can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other, current experiments have found no difference in results regardless of how it is measured:

* *Inertial mass* measures an object's resistance to being accelerated by a force (represented by the relationship *F* = *ma*).
* *Active gravitational mass* measures the gravitational force exerted by an object.
* *Passive gravitational mass* measures the gravitational force exerted on an object in a known gravitational field.

The mass of an object determines its acceleration in the presence of an applied force. The inertia and the inertial mass describe the same properties of physical bodies at the qualitative and quantitative level respectively, by other words, the mass quantitatively describes the inertia. According to Newton's second law of motion, if a body of fixed mass *m* is subjected to a single force *F*, its acceleration *a* is given by *F*/*m*. A body's mass also determines the degree to which it generates or is affected by a gravitational field. If a first body of mass *m*A is placed at a distance *r* (center of mass to center of mass) from a second body of mass *m*B, each body is subject to an attractive force *F*g = *Gm*A*m*B/*r*2, where *G* = 6.67×10−11 N kg−2 m2 is the "universal gravitational constant". This is sometimes referred to as gravitational mass. Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical; since 1915, this observation has been entailed *a priori* in the equivalence principle of general relativity.

* Active gravitational mass is a measure of the strength of an object's gravitational flux (gravitational flux is equal to the surface integral of gravitational field over an enclosing surface). Gravitational field can be measured by allowing a small "test object" to fall freely and measuring its free-fall acceleration. For example, an object in free fall near the Moon is subject to a smaller gravitational field, and hence accelerates more slowly, than the same object would if it were in free fall near the Earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass.
* Passive gravitational mass is a measure of the strength of an object's interaction with a gravitational field. Passive gravitational mass is determined by dividing an object's weight by its free-fall acceleration. Two objects within the same gravitational field will experience the same acceleration; however, the object with a smaller passive gravitational mass will experience a smaller force (less weight) than the object with a larger passive gravitational mass.

### Inertial vs. gravitational mass

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In classical mechanics, Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. That it does is merely an empirical fact.

Albert Einstein developed his general theory of relativity starting with the assumption of the intentionality of correspondence between inertial and passive gravitational mass, and that no experiment will ever detect a difference between them, in essence the equivalence principle.

This particular equivalence often referred to as the "Galilean equivalence principle" or the "weak equivalence principle" has the most important consequence for freely falling objects. Suppose an object has inertial and gravitational masses *m* and *M*, respectively. If the only force acting on the object comes from a gravitational field *g*, the force on the object is:

# F
M
g
.

{\displaystyle F=Mg.}

Given this force, the acceleration of the object can be determined by Newton's second law:

# F
m
a
.

{\displaystyle F=ma.}

Putting these together, the gravitational acceleration is given by:

# a

M
m

g
.

{\displaystyle a={\frac {M}{m}}g.}
This says that the ratio of gravitational to inertial mass of any object is equal to some constant *K* if and only if all objects fall at the same rate in a given gravitational field. This phenomenon is referred to as the "universality of free-fall". In addition, the constant *K* can be taken as 1 by defining our units appropriately.

The first experiments demonstrating the universality of free-fall were—according to scientific 'folklore'—conducted by Galileo obtained by dropping objects from the Leaning Tower of Pisa. This is most likely apocryphal: he is more likely to have performed his experiments with balls rolling down nearly frictionless inclined planes to slow the motion and increase the timing accuracy. Increasingly precise experiments have been performed, such as those performed by Loránd Eötvös, using the torsion balance pendulum, in 1889. As of 2008[update], no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the precision 10−12. More precise experimental efforts are still being carried out.

The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especially friction and air resistance, must be absent or at least negligible. For example, if a hammer and a feather are dropped from the same height through the air on Earth, the feather will take much longer to reach the ground; the feather is not really in *free*-fall because the force of air resistance upwards against the feather is comparable to the downward force of gravity. On the other hand, if the experiment is performed in a vacuum, in which there is no air resistance, the hammer and the feather should hit the ground at exactly the same time (assuming the acceleration of both objects towards each other, and of the ground towards both objects, for its own part, is negligible). This can easily be done in a high school laboratory by dropping the objects in transparent tubes that have the air removed with a vacuum pump. It is even more dramatic when done in an environment that naturally has a vacuum, as David Scott did on the surface of the Moon during Apollo 15.

A stronger version of the equivalence principle, known as the *Einstein equivalence principle* or the *strong equivalence principle*, lies at the heart of the general theory of relativity. Einstein's equivalence principle states that within sufficiently small regions of space-time, it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that the force acting on a massive object caused by a gravitational field is a result of the object's tendency to move in a straight line (in other words its inertia) and should therefore be a function of its inertial mass and the strength of the gravitational field.

## Equivalence principle

By definition of active and passive gravitational mass, the force on

M

1

{\displaystyle M\_{1}}

 due to the gravitational field of

M

0

{\displaystyle M\_{0}}

 is:

F

1

=

M

0

a
c
t

M

1

p
a
s
s

r

2

{\displaystyle F\_{1}={\frac {M\_{0}^{\mathrm {act} }M\_{1}^{\mathrm {pass} }}{r^{2}}}}

Likewise the force on a second object of arbitrary mass2 due to the gravitational field of mass0 is:

F

2

=

M

0

a
c
t

M

2

p
a
s
s

r

2

{\displaystyle F\_{2}={\frac {M\_{0}^{\mathrm {act} }M\_{2}^{\mathrm {pass} }}{r^{2}}}}

By definition of inertial mass:

# F

m

i
n
e
r
t

a

{\displaystyle F=m^{\mathrm {inert} }a}

If

m

1

{\displaystyle m\_{1}}

 and

m

2

{\displaystyle m\_{2}}

 are the same distance

r

{\displaystyle r}

 from

m

0

{\displaystyle m\_{0}}

 then, by the weak equivalence principle, they fall at the same rate (i.e. their accelerations are the same)

a

1

=

F

1

m

1

i
n
e
r
t

=

a

2

=

F

2

m

2

i
n
e
r
t

{\displaystyle a\_{1}={\frac {F\_{1}}{m\_{1}^{\mathrm {inert} }}}=a\_{2}={\frac {F\_{2}}{m\_{2}^{\mathrm {inert} }}}}

Hence:

M

0

a
c
t

M

1

p
a
s
s

r

2

m

1

i
n
e
r
t

=

M

0

a
c
t

M

2

p
a
s
s

r

2

m

2

i
n
e
r
t

{\displaystyle {\frac {M\_{0}^{\mathrm {act} }M\_{1}^{\mathrm {pass} }}{r^{2}m\_{1}^{\mathrm {inert} }}}={\frac {M\_{0}^{\mathrm {act} }M\_{2}^{\mathrm {pass} }}{r^{2}m\_{2}^{\mathrm {inert} }}}}

Therefore:

M

1

p
a
s
s

m

1

i
n
e
r
t

=

M

2

p
a
s
s

m

2

i
n
e
r
t

{\displaystyle {\frac {M\_{1}^{\mathrm {pass} }}{m\_{1}^{\mathrm {inert} }}}={\frac {M\_{2}^{\mathrm {pass} }}{m\_{2}^{\mathrm {inert} }}}}

In other words, passive gravitational mass must be proportional to inertial mass for all objects.

Furthermore, by Newton's third law of motion:

F

1

=

M

0

a
c
t

M

1

p
a
s
s

r

2

{\displaystyle F\_{1}={\frac {M\_{0}^{\mathrm {act} }M\_{1}^{\mathrm {pass} }}{r^{2}}}}

must be equal and opposite to

F

0

=

M

1

a
c
t

M

0

p
a
s
s

r

2

{\displaystyle F\_{0}={\frac {M\_{1}^{\mathrm {act} }M\_{0}^{\mathrm {pass} }}{r^{2}}}}

It follows that:

M

0

a
c
t

M

0

p
a
s
s

=

M

1

a
c
t

M

1

p
a
s
s

{\displaystyle {\frac {M\_{0}^{\mathrm {act} }}{M\_{0}^{\mathrm {pass} }}}={\frac {M\_{1}^{\mathrm {act} }}{M\_{1}^{\mathrm {pass} }}}}

In other words, passive gravitational mass must be proportional to active gravitational mass for all objects.

The dimensionless Eötvös-parameter

η
(
A
,
B
)

{\displaystyle \eta (A,B)}

 is the difference of the ratios of gravitational and inertial masses divided by their average for the two sets of test masses "A" and "B."

η
(
A
,
B
# )
2

(

m

g

m

i

)

A

−

(

m

g

m

i

)

B

(

m

g

m

i

)

A

(

m

g

m

i

)

B

{\displaystyle \eta (A,B)=2{\frac {\left({\frac {m\_{g}}{m\_{i}}}\right)\_{A}-\left({\frac {m\_{g}}{m\_{i}}}\right)\_{B}}{\left({\frac {m\_{g}}{m\_{i}}}\right)\_{A}+\left({\frac {m\_{g}}{m\_{i}}}\right)\_{B}}}}