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

Candidate passages:
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. 
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. 
Conjecture D states that numerical equivalence and homological equivalence agree.
(It implies in particular the latter does not depend on the choice of the Weil cohomology theory).
This conjecture implies the Lefschetz conjecture.
If the Hodge standard conjecture holds, then the Lefschetz conjecture and Conjecture D are equivalent. 
The derivation of Hardy's equations is illustrative.
He begins with a population of genotypes consisting of pure dominants (AA), heterozygotes (Aa), and pure recessives (aa) in the relative proportions p:2q:r with the conditions noted above, that is,Rewriting this as ("p" + "q") + ("q" + "r") = 1 and squaring both sides yields Hardy’s result:  formula_30 Hardy’s equivalence condition is  formula_31 for generations after the first.
Now to prove Hardy's statement: the successor generation's proportions equal, formula_32, if and only if the equivalence condition formula_33 holds.
If formula_32, thenso the equivalence condition holds.
This shows equal proportions imply the equivalence condition.
Now suppose the equivalence condition formula_33 holds.
formula_36 By symmetry, formula_37.
Finally,This shows formula_38, formula_39, and formula_40.
Thus, proportions formula_32 equal.
This shows the equivalence condition implies the proportions equal.
Therefore, the proportions equal if and only if the equivalence condition holds, as was claimed.
Since the equivalence condition always holds for the second generation, all succeeding generations have the same proportions. 
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.