How Is the Strong Equivalence Principle Stronger Than the Weak One?

The strong equivalence principle (SEP) is stronger because it applies not only to ordinary test objects, but also to bodies whose own gravity contributes significantly to their mass. It also places restrictions on local experiments involving gravity itself. The weak equivalence principle (WEP) concerns the universality of free fall for test bodies whose self-gravity is negligible.

This distinction matters when comparing theories of gravity. A theory can make ordinary materials fall at the same rate and still predict different behavior for strongly self-gravitating objects, or allow local gravitational measurements to depend on where or how they are performed.

The weak equivalence principle

In its usual form, the WEP says that two test bodies released at the same place and with the same initial velocity have the same acceleration in a gravitational field, independent of their composition. “Test body” here means an object small enough that its own gravity and its effect on the surrounding field can be neglected.

For example, in a vacuum, a piece of lead and a piece of aluminum should fall with the same acceleration near Earth. The WEP is often expressed as equality between inertial mass and passive gravitational mass: the property determining resistance to acceleration matches the property determining response to an external gravitational field.

The WEP does not, by itself, say that every possible experiment behaves as it would in special relativity in a freely falling laboratory. Nor does its usual test-body formulation settle how a body’s own gravitational binding energy affects its motion.

What the strong principle adds

The terminology is often presented as a hierarchy:

Principle	What it covers
WEP	Universal free fall of test bodies with negligible self-gravity
Einstein equivalence principle (EEP)	WEP, plus local Lorentz invariance and local position invariance for nongravitational experiments
SEP	Extends the EEP’s local-independence idea to gravitational experiments and includes self-gravitating bodies

Local Lorentz invariance means that the outcome of a local nongravitational experiment does not depend on the freely falling laboratory’s velocity. Local position invariance means it does not depend on the laboratory’s location or time. The SEP applies these requirements more broadly, to experiments whose physics involves gravity, and requires universal free fall to extend to bodies with appreciable gravitational self-energy.

“Local” is essential: a sufficiently small freely falling laboratory can eliminate a nearly uniform external gravitational field, but not tidal effects caused by variations in that field across the laboratory. The principles concern the ideal local limit, not the claim that gravity can be eliminated everywhere or over an arbitrarily large region.

A worked comparison: Earth and a neutron star

A rough way to see why self-gravity changes the test is to estimate a body’s gravitational binding energy. For a uniform sphere in Newtonian gravity, its binding energy has magnitude $3GM^2/(5R)$. As a fraction of its rest energy, this is

$$\frac{|U|}{Mc^2} \approx \frac{3GM}{5Rc^2},$$

where $M$ is the body’s mass, $R$ its radius, $G$ the gravitational constant, and $c$ the speed of light.

For Earth, substituting $M \approx 5.97 \times 10^{24}$ kg and $R \approx 6.37 \times 10^6$ m gives a fraction of about $4.2 \times 10^{-10}$. Earth’s gravitational binding energy is tiny compared with its rest energy, so ordinary laboratory tests of free fall are essentially tests of the WEP for bodies with negligible self-gravity.

For a rough neutron-star example, take $M = 1.4$ times the Sun’s mass and $R = 12$ km. The same formula gives a fraction near $0.10$. This is only an order-of-magnitude illustration: a neutron star is relativistic and not a uniform Newtonian sphere. Still, it shows that self-gravity can be a substantial part of a compact body’s energy, unlike for Earth.

A theory might therefore predict that a neutron star and a weakly self-gravitating body respond slightly differently to an external gravitational field, even if it predicts no measurable composition-dependent difference between lead and aluminum. Such a theory could pass ordinary WEP tests while violating the SEP.

How violations are tested

One example is the Nordtvedt effect: if gravitational binding energy contributes differently to a body’s gravitational response and inertial mass, bodies with different self-energy fractions may fall differently toward the same source. The Earth and Moon have different internal structures and self-energy fractions, so lunar laser ranging can test whether they fall identically toward the Sun. This probes an aspect of the SEP that a standard laboratory comparison of materials does not directly isolate.

Other tests examine whether local gravitational measurements depend on the surrounding gravitational environment. General relativity satisfies the SEP in its usual classical formulation, while many alternative theories satisfy the WEP but violate the SEP. The SEP is thus a more demanding condition on a theory, not simply another name for universal free fall.

Common mistakes

• Treating “strong” as a reference to the strong nuclear force. It means the principle has a broader scope, not that it concerns that force.
• Saying the WEP covers every body without qualification. Standard WEP tests use bodies whose own gravity is negligible; SEP tests address significant self-gravity.
• Equating the SEP with the EEP. The EEP concerns local nongravitational experiments; the SEP extends the requirements to gravitational experiments and self-gravitating bodies.
• Assuming freely falling frames remove all gravity. They remove the locally uniform part of a field, but tidal effects remain across a finite region.