The physics of gravity¶

Gravitational redshift¶

Imagine a particle of mass \(m\) dropping through a distance \(h\); the particle starts with energy \(E=m\) and ends with energy \(E=m+mgh\). We then send this back up to the first position, where it has energy \(E'\). We know \(E'=m\), since we’d have a way of making free energy otherwise! Thus

Hence, travelling in a gravitational field causes photons to become redshifted.

Schild’s photons¶

Imagine a photon with frequency \(f\) fired from a point \(A\) to a point \(B\) directly above it in a gravitational field. The photon will be redshifted to a frequency \(f'\), and after some set number of periods \(n\) another photon is sent from a point (in spacetime) \(A'\) to \(B'\). The intervals \(AB\) and \(A'B'\) are the same in all frames, but \(AA'\) and \(BB'\) will vary due to the presence of a gravitaional field.

The falling lift¶

In special relativity an inertial frame is one where Newton’s laws hold, so particles not affected by an external force move in straight lines. Consider the situation of a locally inertial frame, where there are no graviational forces. This occurs in free-fall. Objects in a locally inertial frame stay at rest; if the inertial frame is accelerated then the objects won’t be—they will increase their speed towards the floor of the frame at the same rate as the accleration. This is similar to the observation that all objects fall at the same rate in a gravitational field; Einstein supposed that this was not a coincidence, and postulated that:

Uniform gravitational fields are equivalent to frames that accelerate uniformly relative to inertial frames. (Weak equivalence principle).

Consider a beam of light shining horizontally across an inertial frame; the beam will end up at a point horizontally opposite where it’s emitted, but as the frame is moving in the field in will have moved relative to the torch, and in the frame the beam will appear to be bent by the field.

Tides and geodesic deviation¶

Consider two particles, \(A\) and \(B\), falling towards earth; they start off level at a height \(z(t)\) from the centre of the earth, and separated by a distance \(\xi(t)\). \(z\) and \(\xi\) are proportional, such that

The force on the particle of mass \(m\) at an altitude \(z\) due to gravity is

and so

Thus, as the particles fall they move towards each other.

There is no universal inertial frame¶

In special relativity inertial frames have an infinite extent, but consider two observers free-falling on opposite sides of the earth. Each is inertial, but the second derivative of their mutual separation is non-zero, so special relativity cannot explain what happens in one of the frames relative to the other. Clearly it isn’t then possible for us to, for example, observe matter falling into a black hole and understand it with special relativity.

The theory of General Relativity introduces a principle of general covariance:

All physical laws must be invariant under all coordinate transformations.

Thus the approach of general relativity is to express all physics in a coordinate-independent geometrical description.