8.5 The Intrinsic Signature of Gravity
121
8.5 The Intrinsic Signature of Gravity
Recall our discussion of the equivalence principle. We concluded that because a
uniform gravitational field is equivalent to acceleration of the reference frame it may
be transformed away. Thus the intrinsic signature of gravity is its non-uniformity
and not the presence of a force (Misner 1973) (see Figs. 7.2–7.4). This can now be
seen in a very clear light. If we have a sufficiently large lab and sufficiently accurate
equipment we may be able to detect the difference between gravitational forces in
different parts of the lab. The force difference may be written as
dF
i
= F
i ,k dx
k
= −φ ,i,k dx
k
, signature of gravity: tidal forces.
(8.51)
These are called tidal forces because they do indeed give rise to the tides on earth. That
is, the intrinsic signature of Newtonian gravity is the non-vanishing of the second
derivatives of the potential. But by the correspondence between the Riemann tensor
and the potential derivatives in (8.27) we see that the corresponding signature of the
gravitational field in relativity is
R
α βγ δ = 0, signature of gravity: curved spacetime.
(8.52)
The intrinsic signature of gravity is that the Riemann tensor is nonzero or space is
curved. This is an invariant signature since if the Riemann tensor is nonzero in one
coordinate system it is nonzero in all coordinate systems.
Let us summarize the general relativistic viewpoint on the equivalence principle
and the intrinsic signature of gravity:
1. To the extent that the gravitational field is uniform over some small region
of spacetime it is equivalent to an accelerated system. The gravitational force
may thus be transformed away. This is clearly a local and thus approximate
correspondence.
2. To the extent that the gravitational field varies over the relevant region of spacetime it corresponds to curvature of spacetime. The Reimann tensor field may not
be transformed away. This is clearly a nonlocal and intrinsic characterization of
the gravitational field that distinguishes it from the effects of acceleration.
As we have seen the equivalence principle as stated by Einstein leads to the idea of
a geometric theory of gravity, and to some deep insights and correct predictions such
as the deflection of light by gravity and the redshift of light in a gravitational field. It
has played an important role in the development of general relativity and continues
to elucidate problems concerning electromagnetic effects and quantum effects in
a gravitational field. There is much more that could be said about the equivalence
principle; there are at least 3 versions of it, of which we have only used the first,
usually called the weak equivalence principle (WEP) or the universality of free fall.
The reader is invited to pursue more deeply in the references the other versions and
interpretations, as well as related experimental tests (Will 1993, 2014; Zee 1989).
121
8.5 The Intrinsic Signature of Gravity
Recall our discussion of the equivalence principle. We concluded that because a
uniform gravitational field is equivalent to acceleration of the reference frame it may
be transformed away. Thus the intrinsic signature of gravity is its non-uniformity
and not the presence of a force (Misner 1973) (see Figs. 7.2–7.4). This can now be
seen in a very clear light. If we have a sufficiently large lab and sufficiently accurate
equipment we may be able to detect the difference between gravitational forces in
different parts of the lab. The force difference may be written as
dF
i
= F
i ,k dx
k
= −φ ,i,k dx
k
, signature of gravity: tidal forces.
(8.51)
These are called tidal forces because they do indeed give rise to the tides on earth. That
is, the intrinsic signature of Newtonian gravity is the non-vanishing of the second
derivatives of the potential. But by the correspondence between the Riemann tensor
and the potential derivatives in (8.27) we see that the corresponding signature of the
gravitational field in relativity is
R
α βγ δ = 0, signature of gravity: curved spacetime.
(8.52)
The intrinsic signature of gravity is that the Riemann tensor is nonzero or space is
curved. This is an invariant signature since if the Riemann tensor is nonzero in one
coordinate system it is nonzero in all coordinate systems.
Let us summarize the general relativistic viewpoint on the equivalence principle
and the intrinsic signature of gravity:
1. To the extent that the gravitational field is uniform over some small region
of spacetime it is equivalent to an accelerated system. The gravitational force
may thus be transformed away. This is clearly a local and thus approximate
correspondence.
2. To the extent that the gravitational field varies over the relevant region of spacetime it corresponds to curvature of spacetime. The Reimann tensor field may not
be transformed away. This is clearly a nonlocal and intrinsic characterization of
the gravitational field that distinguishes it from the effects of acceleration.
As we have seen the equivalence principle as stated by Einstein leads to the idea of
a geometric theory of gravity, and to some deep insights and correct predictions such
as the deflection of light by gravity and the redshift of light in a gravitational field. It
has played an important role in the development of general relativity and continues
to elucidate problems concerning electromagnetic effects and quantum effects in
a gravitational field. There is much more that could be said about the equivalence
principle; there are at least 3 versions of it, of which we have only used the first,
usually called the weak equivalence principle (WEP) or the universality of free fall.
The reader is invited to pursue more deeply in the references the other versions and
interpretations, as well as related experimental tests (Will 1993, 2014; Zee 1989).