**Einstein’s general theory of relativity. ** By showing that spatiomaterialism can explain
the truth of Einstein’s special theory of relativity (STR), I have answered
the first part of the Einsteinian reservation about using spatiomaterialism
as the foundation for demonstrating ontologically necessary truths. In this
section, I will answer the second part. Einstein’s general theory of relativity
(GTR) also makes it appear that this is not a spatiomaterial world, and I
will show how its truth can also be explained by spatiomaterialism. .

The way Einstein’s general theory of relativity explains gravitation
does not, at first, seem compatible with spatiomaterialism. The foundation
of the general theory is spacetime, for gravitation is explained as a “curvature”
in spacetime, and since substantivalism about spacetime is incompatible with
substantivalism about space, it seems out of the question that what the general
theory refers to as “curved spacetime” could turn out to be an aspect of space
and matter as substances enduring through time. *Was Einstein Right?)*

It is, however, possible for spatiomaterialism to explain why
Einstein’s general theory of relativity is true. The key is what spacetime
turns out to be in the ontological explanation of the truth of the special
theory of relativity, for that makes it possible to explain curved spacetime
as well. Curved spacetime is also an aspect of space and matter, even though
as substances that endure through time, space and matter exist only at the
present moment.

Though I go on in the next section to suggest an ontological
explanation of quantum mechanics and, in the following section, take up some
basic issues in cosmology, this explanation of the Einstein’s general theory
of relativity pays off the second mortgage that we took out in order to use
spatiomaterialism as the foundation for our philosophical argument. (See Necessary
Truths.) Quantum mechanics is not so crucial to this project, because
there is continuing disagreement about its ontological implications and some
of the possibilities are compatible with spatiomaterialism.

We have already seen how the existence of consciousness can
be explained in a spatiomaterial world (though the unity of consciousness
will not be explained until I take up the mammalian brain in the sixth stage
of evolution), and I have yet to take up the nature of goodness and holiness.
But one of those four mortgages will be repaid when we see that Einsteinian
physics provides not reason for denying that this is a spatiomaterial world.

In fact, spatiomaterialism might welcome the challenge of explaining
Einstein’s general theory of relativity, because that means it does not have
to defend Newton’s theory of gravitation. Newton’s theory is *prima facie
*less hospitable to spatiomaterialism than general relativity. If a force
did act immediately at a distance, it would contradict the principle of local
action, implying that spatiomaterialism is false.

Newton’s theory describes an attractive force by which every
material object acts immediately on every other material object, including
those at a distance. Newton introduced it, in effect, as the best efficient-cause
explanation of Kepler’s laws of planetary orbits, and it was confirmed by
the deduction of many surprising, quantitatively precise predictions of measurements,
becoming the model for the empirical method in physics. Despite its predictive
success, Newton’s law of gravitation had nothing to say about how such forces
are exerted on objects at a distance, except that they act instantaneously
at a distance.

Action at a distance was puzzling to classical physicists, since
it did not fit well with their intuitive understanding of nature as composed
of space and matter in time. Even Newton was uncomfortable with the notion,
and he refused to make any hypotheses about how gravitation worked in his
*Principia*.^{[1]} But action at a distance could not be rejected
for being incompatible with spatiomaterialism, for that would require using
space as an ontological cause, and Newtonian physics did not recognize the
validity of ontological arguments. Still, when Einstein proposed an explanation
of gravitation that implied that gravitational forces propagate at a finite
velocity, even physicists were relieved at not having to believe in action
at a distance. And it did remove what would otherwise be an insuperable objection
to spatiomaterialism.

Einstein’s general theory of relativity was, however, another
highly mathematical hypothesis, which predicted many quantitatively precise
measurements, and since it implies that gravitational acceleration is caused
by a curvature of spacetime, a realist interpretation of Einstein’s theory
seems to imply that spacetime is a substance. But if the real nature of what
exists in addition to mass and energy is spacetime, that is, a four-dimensional
entity in which time is one of the dimensions along with space, then existence
is not in time and “real change” is not ontologically possible. Thus, general
relativity solved one ontological problem, but only by introducing another.
The challenge is, therefore, to explain how curved spacetime can be understood
as an aspect of a world constituted by space and matter as substances that
exist only at the present moment.

**Curved spacetime.
** Having discovered STR by assuming the local equivalence of all
inertial frames, Einstein sought to use the same approach in explaining acceleration
due to gravity, that is, by including reference frames that were being accelerated
by gravitation. Thus, the main assumption of his general theory of relativity
is the equivalence of inertial frames to reference frames falling freely in
gravitational fields.

What Einstein himself called the “principle of equivalence” assumes
that nothing can be detected within any reference frame (that is, locally)
that would distinguish a reference frame in inertial motion from one in free
fall.

Or, to put Einstein’s equivalence principle the opposite way,
a reference frame at rest in a gravitational field is indistinguishable from
one being accelerated by a force; the push that we ordinarily call the “force”
of gravity is actually the force of the earth accelerating us upward from
what is equivalent to inertial motion.

This further equivalence can be only local, however, because
free-falling frames are obviously different in how they are related to the
rest of the world, or globally. Though inertial frames simply continue in
motion indefinitely, free-falling reference frames eventually collide with
the center of gravity, because gravitational fields are imposed by matter
concentrated at certain locations. Thus, what makes the general theory of
relativity general is that it includes both inertial and free-falling reference
frames, and Einstein’s highly mathematical description of how they fit together
as parts of a single world is a theory of acceleration due to gravity.

Einstein’s strategy in GTR paralleled that of his special theory.
In STR, Einstein used his principle of relativity (implying the equivalence
of all inertial frames) to derive a mathematical description of how they must
be related globally (the Lorentz transformation equations). In his general
theory, Einstein started with the assumption that reference frames in free
fall are locally equivalent to inertial frames, and using the four-dimensional,
spacetime mathematics from special relativity, he derived equations describing
how all reference frames, inertial and free-falling frames, are related to
one another. In both theories, the equivalence of reference frames means that
the laws of physics hold the same way on each of them. That means that there
is a mathematical transformation of explanations of events given on any one
reference frame into explanations given on the other in which the laws of
physics have the same form. In special relativity, only a Lorentz transformation
was required, making them Lorentz covariant. But in general relativity, it
is a more general transformation, which includes both inertial frames and
free-fall frames, called “general covariance”. How objects change their motion
depends on centers of mass in their neighborhoods, and using general covariance
as a constraint, Einstein was able to deduce equations that describe what
classical physics attributed to a force of gravity.

Einstein’s general theory of relativity describes a spacetime
world in which the accumulation of matter (both mass and energy) causes a
“curvature” in the surrounding spacetime. This curvature explains the acceleration
that Newtonian physics attributed to a force of gravitation, because it determines,
in turn, the inertial path for any matter located there. (Such an inertial
path though curved spacetime is called a “geodesic”).

GTR also predicts various new phenomena, including the bending
(and slowing down) of light rays passing through gravitational fields, the
precession of the perihelion of Mercury, and a gravitational red shift. These
predictions all differ from classical physics, and since GTR entails the possibility
of black holes, including rotating black holes, it has become the foundation
of cosmology. Except for the precession of Mercury’s perihelion, these phenomena
were not even expected before Einstein’s argument, much less explained, and
so the confirmation of these predictions justified accepting the general theory
by the empirical method of physics.

Realism about the general theory of relativity, like realism
about the special theory, makes it hard to avoid thinking of spacetime as
a substance on a par with what it contains. The curvature of spacetime is
supposed to *cause *the acceleration
of mater that is ordinarily attributed to gravity, and it would be hard to
explain how a property of spacetime can have such an effect on what it contains,
if spacetime did not exist independently of matter.

GTR is, like STR, a highly mathematical theory. Gravitation
is described by the Einstein field equations, which relate the distribution
of mass and non-gravitational energy to the curvature of spacetime. Currently,
GTR is usually interpreted in terms of differential geometry. Spacetime is
postulated as a four-dimensional continuous manifold of points (*M*), and there are two kinds of (tensor)
equations defined everywhere on the manifold. The metric-field tensor (*g*) defines the metric (and geometric) relations
among points in spacetime, and the stress-energy tensor (*T*) represents the distribution of matter
(mass and energy) in spacetime (and its effects).

Jointly, *M, g, *and* T* are called a “model” of GTR, and even
for a world with a particular distribution of mass and energy, there are infinitely
many different, yet empirically equivalent models. They all predict the same
gravitational phenomena, but each model involves a different coordinate system,
for each is based on a different local inertial reference frames at its location
in spacetime, that is, adapted to material objects with different free-fall
trajectories.^{[2]}
Their empirical equivalence is an assumption that Einstein used to derive
his field equations, and it is one of the meanings sometimes given to “general
relativity”. On this geometrical approach, GTR also seems to imply substantivalism
about spacetime, because the four-dimensional manifold of points (*M*)
must be postulated in order to define the metric-field tensor (g) and stress-energy
tensor (T).^{[3]}

The challenge that GTR poses for spatiomaterialism is that it
implies that what exists is spacetime, rather than space and matter existing
as substances in time. In a spacetime ontology, time is another dimension
of what exists on a par with the spatial dimensions (except for a change in
sign and the velocity of light as a scaling factor). Its implications about
time were used in Spatiomaterialism:
Time to show that spatiomaterialism is a better ontological explanation
of nature than spacetime ontology (or “spatiotemporalism”). Substantivalism
about spacetime makes it impossible to explain “real change”, because if what
exists is a four-dimensional entity, and time is part of its structure, then
nothing can be coming into existence or going out of existence as time passes.

As we saw in Spatiomaterialism,
there is no way for spacetime substantivalism to avoid refutation by
the fact that our experience of change itself take place through time and
we are parts of nature, except by postulating an additional, subjective substance,
for whom spacetime and the events it contains have the appearance of real
change. Not only does the addition of such a subjective substance make spacetime
ontology more complex, but it also poses the problem of relating eternal and
enduring substances as parts of the same world, a problem that Plato never
solved. And even if it could be solved, this modification would be *ad hoc*, for it would explain nothing but
the *appearance *that change takes place through time. There is, therefore,
no question that spatiomaterialism is a better ontology, if it is possible.

In order to show that spatiomaterialism is possible is to show
that it can explain why GTR appears to be true, and that means explaining
all the relevant phenomena on the assumption that nothing exists but space
and matter enduring through time. This is to describe a model or solution
of the Einstein field equations that differs from the prevailing geometrical
interpretation because, instead of postulating a four-dimensional manifold
and defining geometrical objects on it, spatiomaterialism postulates space
and matter as substances enduring through time. Nothing exists in a spatiomaterial
world but what exists at present, and thus, the interaction of space and matter
must somehow have an aspect that explains what Einsteinians are referring
to when they talk about “curved spacetime” and that aspect must explain all
the phenomena predicted by the general theory.

We can tell that not in principle impossible for a world of
substances that exist only at the present moment to explain the truth of GTR,
because even on the received geometrical interpretation, there is a standard
of simultaneity implicit in each model’s assignment of space and time coordinates
to every event in the universe. All the spacetime events with the same temporal
coordinates that we now have in some model for our universe (a certain “simultaneity
hypersurface” in curved spacetime) *could *be all that actually exists at the present moment, and their
spatial coordinates *could *be referring
to parts of a three dimensional Euclidean substance. Of course, this could
be true of only one model, for although every model assigns some coordinates
to us now, different models entail different standards of simultaneity, and
if different models were *ontologically
*equivalent, the substances constituting the world would have to include
spacetime.

Moreover, in order to hold, in effect, that one of all possible
models represents absolute space and time, spatiomaterialism would have to
show that there is a law of gravitation that explains not only the approximate
truth of the Newtonian theory in it, but also all the new phenomena predicted
by GTR. We can also tell that such a law is not in principle impossible, because
GTR itself implies that the relevant events in that model are all related
in a regular way. Still, the regularity would have to be described without
referring to spacetime or spacetime curvature, that is, explained as constituted
by (Euclidean) space and matter enduring through time. And there would be
problem about the regularity, only if its description turned out to be very
complex.

Finally, since spatiomaterialism would take reality to be equivalent
to what exists in a single model of GTR according to the received geometrical
approach, we should also expect the spatiomaterialist law of gravitation to
explain why different models are observationally equivalent, that is, to explain
“general relativity”, in the sense that enabled Einstein to derive his mathematical
representation of gravitation.

This is a tall order, but it is possible, as I will show here
by giving an ontological explanation of why Einstein’s GTR is true. It is
an intuitively intelligible explanation, rather than a mathematical explanation,
because what is required to explain the truth of any theory ontologically
is showing that there are aspects of the substances postulated by the ontology
that correspond to the theory. That requires a qualitative argument, which
identifies the kinds of regularities and how they are related according to
the theory, and then shows that they can all correspond to aspects of the
same world. To be sure, the aspects of the substances pointed out must be
quantitatively adequate as well. But that is rather trivial, once the qualitative
argument has shown what the parameters are, how they are related to one another,
and the signs and order of magnitude of their quantities, because substances
can be postulated as having whatever quantitative aspects are required to
make the measurements come out correctly. Thus, I will leave it as a challenge
to those who would disprove spatiomaterialism to show that the aspects identified
here cannot all be quantitatively accurate.

^{[1]}
Newton later suggested various mechanisms to account for gravitation. See
Burtt
(1980, pp. 264ff).

^{[2]}
This equivalence can also be put mathematically, as Hoefer
(1996) does: “By taking one model <*M, g, T*> and applying a diffeomorphism
*h* (essentially, a permutation
of the points in *M* satisfying
certain restrictions), one can generate a ‘new’ model of the theory <*M,
g, T*> which is qualitatively identical, but which has the material
contents and the metric field distributed differently over the point manifold
of *M*” (7-8).

^{[3]}
This is the orthodox approach, represented by Michael Friedman
(1983), and John Earman
(1989). But Earman and John Norton use the “hole argument” to raise doubts
about the four-dimensional manifold of points being a substance are raised
by the “hole argument”. See Earman
and Norton (1987). But spacetime substantivalism has its defenders,
such as Hoefer (1996). Though the spatiomaterialist theory does not need
to answer the hole argument to defend its substantivalism about space, it
may be relevant to mention that it sees the “hole argument” as an artifact
of the mathematical formulation of GTR. Instead of seeing the models as
different (locally) inertial frames used to assign coordinates throughout
the universe with a certain standard of simultaneity, the hole argument
interprets their observational equivalence as a mere mathematical operation
(a diffeomorphism; see previous footnote), and that makes it possible to
hold that there can be “holes”, or regions where, in effect, different standards
of simultaneity hold. The spatiomaterialist ontological explanation of the
observational equivalence of different models of GTR will be given at the
end of this explanation of the general theory itself.