Clock Hypothesis and Observer in General Relativity

Figure 1: Spacetime diagram representing the Clock Hypothesis. The tick rate of a clock along an accelerated worldline (cyan curve) as measured by coordinate time is given by \(d\tau = dt/\gamma(t)\), independent of its acceleration. (AI-generated illustration)

In relativity theory, understanding the accelerated observer is very important. In this discussion, I want to emphasize the significance of the clock hypothesis, which, along with Einstein's equivalence principle, dictates the flow of time for a particle following a trajectory in curved spacetime. I have also proved the existence and uniqueness of a geodesic (i.e., a locally freely falling frame) and the concept of Gaussian normal coordinates. In the last section, I have discussed moving reference frames in GR, and therefore, the idea of simultaneity in GR.

Statement of Clock Hypothesis

The clock hypothesis states that a clock's ticking rate matches its Momentarily Comoving Reference Frame (MCRF) at every instant, even during acceleration in curved spacetime.

1. Accelerating Observer and the Clock Hypothesis

Consider an observer S accelerating with a proper acceleration g with respect to an inertial observer O. For simplicity, we assume there is no influence of gravity at any point in spacetime.

First, we define proper acceleration: it is the acceleration felt directly by the observer S. Mathematically, an instantaneously co-moving inertial frame to S will measure this constant acceleration g.

Secondly, we address how fast S's clock ticks within the coordinate frame of the inertial observer O. As measured by O, at any coordinate time t, the observer S has an instantaneous velocity v(t) and acceleration g. The clock hypothesis asserts that the rate at which a clock ticks in the accelerated frame S, as measured by the inertial observer O, depends only on its instantaneous velocity v and is independent of its acceleration.

Consequently, the rate of S's clock ticking as measured by O is inversely proportional to the Lorentz factor:

If \(d\tau\) is the infinitesimal proper time elapsed on S's clock while a coordinate time interval \(dt\) has evolved in O's frame, the clock hypothesis ensures that:

Notice that there is explicitely no g dependence in this relationship. Integrating this expression yields the total proper time elapsed:

This integral is identically the spacetime interval evaluated between the starting and final events in Minkowski space. In a nutshell, by constructing an instantaneous inertial frame and applying the clock hypothesis, we recover the elapsed proper time for any accelerated trajectory.

2. Particle Following a General Trajectory in Spacetime

Now, let us extend this formulation to a particle following an arbitrary, potentially non-geodesic trajectory in a general curved spacetime, parameterized by \(x^\mu(\tau)\).

Consider a freely-falling co-moving frame (or a co-moving geodesic) at a specific point \(x^\mu(\tau_0)\) along the trajectory, which we denote as frame Q. The mathematical existence and uniqueness theorem for geodesic differential equations guarantees that there always exists a unique geodesic satisfying any prescribed initial conditions of position and four-velocity. (Look Appendix for complete mathematical argument)

Proof. Although it's a founding assumption of The Einstein Equivalence Principle, one can mathematically prove it using Gaussian-normal coordinates, i.e. a local coordinate system along the geodesic. In this frame, geodesics crossing the point are all radial, from which we can deduce that Christoffel symbols are zero, which implies locally first derivatives of metric tensor are zero. Therefore, under another suitable choice of orthonormal basis (tetrads) at the point, one can deduce locally flat Minkowski metric.

In this local frame Q, the particle is instantaneously stationary (\(v = 0\)). By the Einstein Equivalence Principle, this configuration is locally physically equivalent to a particle accelerating in an accelerated lift (to account for particle's motion in background curved spacetime metric) and co-moving inertial frame setup analyzed in Section 1. Therefore, invoking the clock hypothesis, the rate of clock ticking in the particle's own frame matches that of the locally co-moving flat inertial frame Q, whose metric is flat at the point.

The infinitesimal time elapsed in Q's frame satisfies:

where \(ds\) is the invariant spacetime interval. Consequently, the clock carried by the physical particle measures the accumulated spacetime interval along the path traced by that particle through spacetime. The general mathematical statement equivalent to the clock hypothesis is written as:

This result shows that the physical clock attached to the particle measures the spacetime interval traversed by the particle. This result will be used in determing the coordinate transformations, when one switches to an accelerating frame of reference.

Example: Gravitational Time Dilation Derivation

Solution:

Consider two events at the first spatial location: \[ E_1 = (x_1, y_1, z_1, t_1) \quad \text{and} \quad E_2 = (x_1, y_1, z_1, t_1 + dt_1) \]

At Event 1 and Event 2, a light ray is flashed from \((x_1, y_1, z_1)\) towards \((x_2, y_2, z_2)\). Let the light ray emitted at Event 1 reach \((x_2, y_2, z_2)\) at time \(t_2\). Then the light ray emitted at Event 2 reaches \((x_2, y_2, z_2)\) at time \(t_2 + dt_2\).

Since the metric components are time-independent, the coordinate time interval for the light travel is constant, i.e.: \[ dt_2 = dt_1 \]

The proper time elapsed in particle 1's frame between the flashes is given by: \[ d\tau_1 = \sqrt{g_{00}(x_1, y_1, z_1)} \, dt_1 \]

And the proper time elapsed in particle 2's frame between the flashes is: \[ d\tau_2 = \sqrt{g_{00}(x_2, y_2, z_2)} \, dt_2 \]

Therefore, the ratio of the clock ticking rates is:

Now, if we take the second point to spatial infinity, \((x_2, y_2, z_2) \to \infty\), the metric approaches the flat space metric, so \(g_{00} \to \eta_{00}\). This implies:

For the case of the Schwarzschild metric: \[ g_{00} = 1 - \frac{2GM}{r} \]

Therefore, the clock ticking rate at coordinates \((r, \theta, \phi)\) is:

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Appendix:

Appendix 1: Local Existence and Uniqueness of Geodesics

In general relativity, a geodesic represents the path of a freely falling particle, generalizing the concept of a straight line to curved spacetime. Here I will establish the local existence and uniqueness of geodesics. For a semi-Riemannian manifold \((M, g)\), a curve \(\gamma: I \to M\) parameterized by proper time \(\tau\) is a geodesic if its tangent vector field \(\dot{\gamma} = \frac{d\gamma}{d\tau}\) is parallelly transported along the curve itself. In a local coordinate chart \(x^\mu\), this condition yields the geodesic equation:

where \(\Gamma^\mu_{\alpha\beta}\) are the Christoffel symbols of the second kind, defined in terms of the metric tensor \(g_{\mu\nu}\) and its inverse \(g^{\mu\nu}\) as:

Notice that geodesic equation is coordinate invariant, i.e. it takes the same form under any coordinate transformation, but is not invariant under reparametrization. It is invariant only under affine parametrisation of proper time \(\tau\).

The geodesic equation is a system of \(n\) second-order, non-linear ordinary differential equations (ODEs), where \(n\) is the dimension of the manifold (e.g., \(n=4\) for spacetime).

To prove local existence and uniqueness, we reduce this system to a first-order system of \(2n\) equations. Let us introduce a different set of variables \(y^\mu = \frac{dx^\mu}{d\tau}\) representing the components of the velocity vector.

Using these variables, the geodesic equation can be written as:

We can express this system in vector form. Let \(z = (x, y) = (x^1, \dots, x^n, y^1, \dots, y^n) \in U \times \mathbb{R}^n\) where \(U \subseteq \mathbb{R}^n\) is a coordinate chart on \(M\). The system becomes:

with components \(F^\mu(x, y) = y^\mu\) for \(1 \leq \mu \leq n\), and \(F^{n+\mu}(x, y) = -\Gamma^\mu_{\alpha\beta}(x) y^\alpha y^\beta\) for \(1 \leq \mu \leq n\).

To establish existence and uniqueness of solutions, we can apply the Picard–Lindelöf theorem for this first-order differential equations.

One can check that the Picard–Lindelöf theorem can be applied to this problem, assuming metric components to be smooth functions of coordinates. Therefore, we can state the local existence and uniqueness of geodesics as follows:

This fundamental result guarantees that, locally, a freely falling observer's path through spacetime exists and is completely and uniquely determined by their starting position and initial velocity vector.

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References and Further Reading