Download Gravitomagnetism - Stanford University

EPGY Summer Institute
Special and General Relativity 2012
Gravitomagnetism
It is often noticed that the Newtonian relation for gravity bares a striking resemblance to that of electrostatics, i.e.
Coulomb’s law,
F~
=
−
kQq
F~ = 2 r̂.
r
GM m
r̂
r2
It does turn out this similarity is only skin deep, however some things can be learned by making analogies. Here
we will explore a phenomena that was thought of, but never measured until very recently -gravitomagnetism. In
fact, any type of force that has monopole charges (e.g. point masses and point electric charges) will have some type
of magnetic phenomena associated with it. It is a requirement of relativity that such a phenomena exist. So let’s
see what we might make out of the gravitational analogue of magnetism. Here we will do an entirely Newtonian
exploration, which will need to be corrected by general relativity. You might ask yourself, ‘why haven’t I studied
gravitomagnetism in any of my physics courses?’ The answer to this will be seen when we get our result.
Review electric and magnetic forces
To begin, let’s recall the basics of electromagnetism. We start with Coulomb’s law written in a more appropriate
manner,
F~ =
1 Qq
r̂,
4πǫ0 r2
where ǫ0 = is the permittivity of free space. The Lorentz force law gives the force resulting from an electric charge
~
moving through a magnetic field, B,
F~
=
~
q~v × B.
The direction of the force is given by the right hand rule1 Consider the simple example of a B field created by a
single line current. The magnitude of the field is,
B
=
µ0
I
,
2πr
where µ0 = is the permeability of free space. We can insert this into the Lorentz force law (ignoring directions ~ are perpendicular),
consider magnitude only when ~v and B
F
=
qvµ0
I
.
2πr
Lastly, recall Maxwell’s electromagnetic wave equation (for a one-dimensional wave traveling in the x direction),
∂2E
1 ∂2E
−
2
∂t
ǫ0 µ0 ∂x2
= 0.
It was mentioned how the coefficient of the second term turns out to be close to the speed of light squared. Thus we
have the relation c = √ǫ10 µ0 .
On to gravity
Now we make an analog to magnetism but pertaining to gravity. We will also borrow one fact from general relativity
-the force of gravity, like E&M, is mediated at the speed of light. We will introduce a new quantity BG , the
gravitomagnetic field and assume the relations are the same,
F
=
GM m
r2
~ G.
F~ = m~v × B
In the second expression we note that mass is the charge of gravity, thus when a mass moves through a gravitomagnetic
field it experiences a Lorentz-like force. We can reexpress Newton’s universal law in terms of a permittivity associated
to gravity. That is,
F
=
1 Mm
GM m
=
2
r
4πǫG r2
−→
ǫG =
1
1
=
= 1.2 × 109 kg 2 /N m2
−11
4πG
4π(6.67 × 10 N m2 /kg 2 )
~ =A
~ × B.
~ To find the direction of C,
~ given the direction of A
~ and B,
~ point the fingers of your right
the cross product C
~ and rotate, palm first, through the acute angle to the second vector B.
~ Your thumb points in
hand in the direction of the first vector A
~ The order of the two vectors is important.
the direction of the resultant, C.
1 Consider
We imagine a mass current creating a gravitomagnetic field just as above. The gravitomagnetic field should be
expressible as,
BG
= µG
Im
,
r
where Im is the mass current measured in kg/s. As before, we insert this into the Lorentz force law for gravitomagnetism,
F = mvµ0
Im
.
r
The only unknown is the permeability for gravitomagnetism. To find its value we assume a similar wave equation
as Maxwell’s and note that gravity should be mediated at the speed of light as well. This gives us,
c
=
1
√
ǫ G µG
−→
µG =
1
= 9.3 × 10−27 m/kgs.
ǫ G c2
Discussion and example
It might become clear now why gravitomagnetism is never discussed in basic physics courses. The effect is extremely
weak. It is also known that to properly describe it, you need to consider general relativity. Our exercise here was
only to emphasize that it is not easily measurable. In fact, the first measurement was performed by Gravity Probe
B in 2005, and that experiment took 40 years to construct!
Just to bring home how weak this force is, let’s concoct a system and see what the magnitude of the gravitomagnetic force would be. Consider a long line of 747s, one after another and flying at 250 m/s. Take 400,000 kg as the
mass of a 747 which is about 70 m in length. The mass current is then Im = mv/l = (400000 kg)(250m/s)/(70m) =
1.4 × 106 kg/s. The gravitomagnetic field is thus,
BG
= µG
Im
= (1.33 × 10−20 1/s)/r.
r
Let’s have another 747 fly parallel to this line at the same speed. It will represent a point mass moving through
this gravitomagnetic field. (Clearly this is an exaggeration, however we only need order of magnitude estimates.) In
order to maximize the force, let’s have it fly 1 meter from the mass current. Thus the gravitomagnetic force on this
747 is,
F
=
mvBG = (400, 000 kg)(250 m/s)(1.3 × 10−20 1/s) = 1.3 × 10−12 N
This is an extremely small force to detect. The gravitational (gravitoelectric, if you will) force between two 747s
that are 1 m apart is about 10 N. (Of course, this all assumes that 747s are point-like objects, they clearly are not.)