Question: What does gravity influence, assuming the equivalence principle holds?
About the article "General relativity", section "Gravitational time dilation and frequency shift".

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The theory of quantum field theory in curved spacetime can be considered as a first approximation to quantum gravity.
A second step towards that theory would be semiclassical gravity, which would include the influence of particles created by a strong gravitational field on the spacetime (which is still considered classical and the equivalence principle still holds).
The reason is that gravity is not renormalizable in QFT. 
Assuming that the equivalence principle holds, gravity influences the passage of time.
Light sent down into a gravity well is blueshifted, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted; collectively, these two effects are known as the gravitational frequency shift.
More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.
Gravitational redshift has been measured in the laboratory and using astronomical observations.
Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks, while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS).
Tests in stronger gravitational fields are provided by the observation of binary pulsars.
All results are in agreement with general relativity.
However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.
The strong equivalence principle suggests the laws of gravitation are independent of velocity and location.
In particular, "The gravitational motion of a small test body depends only on its initial position in spacetime and velocity, and not on its constitution."
and  "The outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime."
The first part is a version of the weak equivalence principle that applies to objects that exert a gravitational force on themselves, such as stars, planets, black holes or Cavendish experiments.
The second part is the Einstein equivalence principle (with the same definition of "local"), restated to allow gravitational experiments and self-gravitating bodies.
The freely-falling object or laboratory, however, must still be small, so that tidal forces may be neglected (hence "local experiment").
This is the only form of the equivalence principle that applies to self-gravitating objects (such as stars), which have substantial internal gravitational interactions.
It requires that the gravitational constant be the same everywhere in the universe and is incompatible with a fifth force.
It is much more restrictive than the Einstein equivalence principle.
The strong equivalence principle suggests that gravity is entirely geometrical by nature (that is, the metric alone determines the effect of gravity) and does not have any extra fields associated with it.
If an observer measures a patch of space to be flat, then the strong equivalence principle suggests that it is absolutely equivalent to any other patch of flat space elsewhere in the universe.
Einstein's theory of general relativity (including the cosmological constant) is thought to be the only theory of gravity that satisfies the strong equivalence principle.
A number of alternative theories, such as Brans–Dicke theory, satisfy only the Einstein equivalence principle. 
The Weak Equivalence Principle says that all objects fall the same way in a gravity field, no matter what they are made of.
The Earth and the Moon have very different compositions—for example, the Earth has a large iron core, but the Moon does not.
Furthermore, both are in orbit around the Sun, meaning they are both falling towards the Sun at all times, even as they revolve around each other.
If the Earth and the Moon were affected differently by the gravity of the Sun, this would directly affect the orbit of the Moon around the Earth.
But as closely as scientists can measure, the orbit of the Moon is just as predicted from assuming that gravity acts the same on each — to within 1 part in , the Earth and the Moon fall towards the Sun in exactly the same way, despite their different compositions.
APOLLO will lead to even tighter limits.
What about the Strong Equivalence Principle?
According to Einstein's General Relativity, the mass of any object consists of two parts—the mass of the atoms themselves, plus the mass of the energy that holds the object together.
The question is whether the energy portion of mass behaves like the traditional part—does it contribute to measured gravity of the object?
To the inertia?
In General Relativity, the self energy affects both the gravity field and inertia, and does so equally.
This is the Strong Equivalence Principle (SEP).
Other modern theories, such as string theory, quintessence, and various forms of quantum gravity, almost all predict a violation of the Strong Equivalence Principle at some level.
Additionally, many puzzling experimental results, such as Galaxy rotation curves that imply dark matter or supernova observations that imply dark energy, could also potentially be explained by alternative theories of gravity (see, for example, MOND).
Therefore experimentalists believe it is important to make the most precise measurements of gravity possible, looking for any possible anomalies or confirming Einstein's predictions.
Precise ranging to the Moon can test the SEP since the Earth and the Moon have a different fraction of their mass in the energy component.
Precision measurements are needed since this component is very small—if mE is the self energy of the Earth — the energy needed to spread the atoms of the Earth out to infinity against the attraction of gravity — then the mass of the Earth is "decreased" by about mE/c2 = of the Earth’s total mass.
The self energy of the Moon is smaller yet, about of its mass.
(The contribution for any object of laboratory size is negligible, about , so only measurements of planet-sized or bigger objects have any hope of seeing this effect.
)If the Moon just revolved around the Earth, there would be no way to tell what fraction of the Moon's or the Earth's gravity was caused by each form of mass, since only the total can be measured.
However, the orbit of the Moon is also strongly affected by the gravity of the sun—in essence, Earth and Moon are in freefall around the sun.
If the energy portion of mass behaves differently than the conventional portion, then the Earth and the Moon will fall differently toward the Sun, and the orbit of the Moon around the Earth will be affected.
For example, suppose the energy part of the mass does affect gravity, but does not affect inertia.
Then:From our perspective on Earth, this would appear as a displacement, or polarization, of the lunarorbit away from the sun with an amplitude of 13 meters.
If the violation went the other way, with theself energy possessing inertial mass but not gravitational mass, the lunar orbit would appear to bepolarized toward the sun by the same amplitude.
The calculation of the amplitude is complicated, but a crude estimate may be derived by multiplying the Earth’s orbital radius of The signature of an EP violation is very simple, depending only on the distance of the Moon from the Sun.
This repeats about every 29.5 days, somewhat longer than the time the Moon takes to go around the Earth once, which is 27.3 days.
(This difference arises since the Earth moves along its orbit as the Moon goes around, so the Moon has to make a little more than one orbit to get back to the same position relative to the Sun.)
This makes EP particularly easy to measure, since many confounding effects such as tides or weather will not repeat at 29.5 day intervals.
Unfortunately, there is one effect—radiation pressure acting on the orbit of the Moon—that does repeat each 29.5 days.
Fortunately, it is small, less than 4 mm, and fairly easy to model so it can be subtracted out.
Finally, even if the experiments show no effect, there is a tiny theoretical loophole.
The measurements show the sum of the WEP and SEP violations.
If the experiments show no effect, the most natural explanation is that neither WEP or SEP are violated.
But it is conceptually possible that both are violated, and by equal and opposite amounts.
This would be an incredible coincidence since WEP and SEP depend on very different and arbitrary properties—the exact composition of the Earth and the Moon, and their self-energies.
But this unlikely case cannot be completely ruled out until either other solar system bodies are measured to similar precision, or laboratory experiments reduce the bounds on WEP violations alone. 
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics.
In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics.
(The defining symmetry of special relativity is the Poincaré group which also includes translations and rotations.)
The differences between the two become significant when we are dealing with speeds approaching the speed of light, and with high-energy phenomena.
With Lorentz symmetry, additional structures come into play.
They are defined by the set of light cones (see the image on the left).
The light-cones define a causal structure: for each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image).
These sets are observer-independent.
In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the space–time's semi-Riemannian metric, at least up to a positive scalar factor.
In mathematical terms, this defines a conformal structure.
Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected.
Bringing gravity into play, and assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames.
Instead there are approximate inertial frames moving alongside freely falling particles.
Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.
A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames.
But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf.
below).
The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.
The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity.
The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity.
In the language of spacetime geometry, it is not measured by the Minkowski metric.
As in the Newtonian case, this is suggestive of a more general geometry.
At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian.
Consequently, we are now dealing with a curved generalization of Minkowski space.
The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric.
Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).