Equivalence Principle
Constant Relative Acceleration
Linear to Curvilinear Transformations
Gravitational Deflection of Light
Following the formulation of special relativity in 1905, which unified physics under the postulate of a constant speed of light in all inertial frames, attention turned towards a more comprehensive theory of relativity.
In the subsequent decade Einstein decade aimed to generalize the principle of relativity to encompass reference frames in arbitrary states of motion, specifically those undergoing acceleration with respect to one another.
An inertial frame is defined as a system of coordinates in which Newton’s first law holds; that is, an object with no net force acting upon it moves with constant velocity. The extension to non-inertial, or accelerated, frames was done with the theory of general relativity in 1915.
A pillar of this theory is the equivalence principle, which posits a connection between gravitation and acceleration.
To explore this principle, we will analyze a thought experiment involving an elevator.
Let there be two reference frames: a frame S, with coordinates (x, y, z), considered fixed with respect to the Earth and assumed to be inertial, and a frame S^\prime, with coordinates (x^\prime, y^\prime, z^\prime), which is moving with the elevator.
We will operate within the non-relativistic limit, where the speed of light c is taken to be infinitely large.
Let the motion be constrained to the vertical direction. The coordinate systems are aligned such that at the initial time t=0, the origins coincide, z = z^\prime = 0.
The position of the elevator’s floor in the fixed frame S is described by a function L(t). The relationship between the vertical coordinates in the two frames is given by the transformation:
z^\prime = z - L(t)
Constant relative velocity
First, let us examine the scenario where the elevator ascends with a constant velocity \mathbf v = v \mathbf k.
In this configuration, both S and S^\prime are inertial frames. The position of the elevator floor is a linear function of time:
L(t) = vt
The Galilean transformation relating the coordinates of a point P in S to its coordinates in S^\prime is:
\begin{aligned} t^\prime &= t \\ x^\prime &= x \\ y^\prime &= y \\ z^\prime &= z - vt \end{aligned}
Consider a particle of mass m subjected to a force \mathbf F = F_z \mathbf k in the fixed frame S. According to Newton’s second law, its equation of motion is:
\mathbf F = m \frac{\mathrm d^2 \mathbf r}{\mathrm d t^2}
In terms of the vertical component, this becomes:
F_z = m \frac{\mathrm d^2 z}{\mathrm d t^2} = m \ddot z
To determine the form of this law in the moving frame S^\prime, we must transform the acceleration. Taking the second time derivative of the coordinate transformation for z^\prime yields:
\begin{aligned} & \frac{\mathrm d z^\prime}{\mathrm d t} = \frac{\mathrm d z}{\mathrm d t} - v \\ & \frac{\mathrm d^2 z^\prime}{\mathrm d t^2} = \frac{\mathrm d^2 z}{\mathrm d t^2} \end{aligned}
This shows that the acceleration is invariant under this transformation, \ddot z^\prime = \ddot z. The equation of motion in the S^\prime frame, described by an observer within the elevator, is:
F_z^\prime = m \ddot z^\prime
Given that the accelerations are identical, and assuming the force measured is the same (F_z^\prime = F_z), the mathematical form of Newton’s second law is preserved.
This invariance is a defining characteristic of transformations between inertial reference frames.
Constant relative acceleration
Next, we consider the elevator moving with a constant upward acceleration \mathbf a = g \mathbf k. The frame S^\prime is now a non-inertial frame.
The position of the elevator floor is given by:
L(t) = \frac{1}{2}gt^2
The coordinate transformation between the inertial frame S and the accelerated frame S^\prime is:
\begin{aligned} t^\prime &= t \\ x^\prime &= x \\ y^\prime &= y \\ z^\prime &= z - \frac{1}{2}gt^2 \end{aligned}
The equation of motion for a particle of mass m in the inertial frame S remains:
F_z = m \ddot z
We again find the acceleration in the S^\prime frame by differentiating the transformation:
\begin{aligned} & \frac{\mathrm d z^\prime}{\mathrm d t} = \frac{\mathrm d z}{\mathrm d t} - gt \\ & \frac{\mathrm d^2 z^\prime}{\mathrm d t^2} = \frac{\mathrm d^2 z}{\mathrm d t^2} - g \end{aligned}
Substituting \ddot z = F_z/m into the expression for \ddot z^\prime, we obtain the equation of motion as perceived by an observer in the accelerating frame:
m \ddot z^\prime = m (\ddot z - g) = F_z - mg
The observer in the elevator measures an effective force, F_z^\prime, which is not equal to the true force F_z. The equation of motion in S^\prime is:
F_z^\prime = m \ddot z^\prime = F_z - mg
An additional term, -mg, has emerged in the description of motion. This term, which arises purely from the non-inertial nature of the reference frame, is termed a fictitious force.
It is indistinguishable in its mathematical form from the force of gravity experienced by the mass m at the surface of the Earth.
The effects of being in a uniformly accelerating reference frame are locally identical to the effects of being in a uniform gravitational field.
The physical manifestation of gravity, the gravitational force, is proportional to the mass of the object.
This mass, known as the gravitational mass, appears to be identical to the inertial mass that appears in Newton’s second law.
This empirical fact, the equality of inertial and gravitational mass, is what allows the fictitious force due to acceleration to mimic gravity so perfectly.
An observer confined to the elevator, feeling a downward force on all objects, cannot perform any local experiment to distinguish between two possibilities:
- the elevator is at rest in a uniform gravitational field of strength g,
- the elevator is in a region devoid of gravity but is accelerating uniformly at a rate g.
The fact that they are not distinguishable is the essence of the equivalence principle.
Linear to curvilinear transformations
The relationship between different reference frames can be understood through coordinate transformations.
In the context of non-relativistic mechanics and frames in uniform relative motion, the transformation is linear.
For the case of an elevator moving with constant velocity v, the coordinates (t, z) of an event in the stationary frame S are related to the coordinates (t^\prime, z^\prime) in the moving frame S^\prime by:
\begin{aligned} t^\prime &= t \\ z^\prime &= z - vt \end{aligned}
The horizontal coordinates (x, y) remain unchanged. This transformation, a Galilean boost, maps straight lines in the (t, z) spacetime diagram to other straight lines.
This geometric property has a direct physical correspondence. In Newtonian physics, a free particle, one not subject to external forces, traverses a straight-line path at a constant velocity within an inertial frame.
A linear coordinate transformation preserves this rectilinear motion, which is consistent with the fact that a frame moving at a constant velocity relative to an inertial frame is also inertial.
The situation changes when we consider transformations to an accelerated frame. For an elevator with constant acceleration g, the transformation becomes non-linear:
z^\prime = z - \frac{1}{2}gt^2
This is an example of a curvilinear coordinate transformation.
Under this mapping, a straight line in the S frame (for instance, the trajectory of a free particle, z(t) = z_0 + v_0 t) is transformed into a curved line in the S^\prime frame.
An observer in S^\prime would describe the particle’s motion as accelerated.
Einstein’s insight was to identify this mathematical feature, the emergence of apparent acceleration through curvilinear coordinate transformations, with the physical phenomenon of gravitation.
The framework of special relativity is built upon linear Lorentz transformations, which preserve the form of physical laws between inertial frames.
To incorporate gravity, it became necessary to consider a broader class of transformations, the curvilinear transformations of spacetime itself.
Gravitational deflection of light
A direct and startling consequence of the equivalence principle concerns the behavior of light in a gravitational field. Prior to Einstein’s work, the prevailing scientific view held that light, being massless, was unaffected by gravity.
The equivalence principle, however, necessitates a different conclusion. If the effects of a uniform gravitational field are indistinguishable from those of uniform acceleration, and if acceleration affects the path of light, then gravity must also affect the path of light.
To see this, we return to our thought experiment.
Consider a beam of light emitted horizontally from one side of the elevator at time t=0. In the inertial frame S, light propagates in a straight line at speed c. Its trajectory can be described by the parametric equations:
\begin{aligned} x(t) &= ct \\ z(t) &= h_0 \end{aligned}
where h_0 is the initial height of emission. For simplicity, let us set the coordinate origin such that h_0 = 0.
Now, let us determine the trajectory of this light beam as observed from within the accelerating frame S^\prime. We apply the coordinate transformation:
\begin{aligned} x^\prime &= x = ct \\ z^\prime &= z - \frac{1}{2}gt^2 = 0 - \frac{1}{2}gt^2 \end{aligned}
We can express the path of light in the S^\prime frame by eliminating the time parameter t. From the first equation, we have t = x^\prime/c. Substituting this into the second equation yields the spatial trajectory:
z^\prime = -\frac{1}{2}g \left(\frac{x^\prime}{c}\right)^2 = -\frac{g}{2c^2} (x^\prime)^2
This is the equation of a parabola. An observer inside the accelerating elevator will see the beam of light follow a curved path, bending downwards towards the floor, just as a massive particle would.
By the equivalence principle, an observer in an elevator at rest in a uniform gravitational field must observe the exact same phenomenon.
Therefore, Einstein concluded that a gravitational field must bend the trajectory of a light ray. The use of curvilinear coordinate transformations to describe accelerated frames results in a modification of the laws of physics, introducing apparent forces.
The equivalence principle change this observation to a physical statement: these apparent gravitational fields, which materialize from the geometry of the coordinate system, are physically indistinguishable from gravitational fields generated by mass and energy.
Gravity, in this view, ceases to be a force and becomes a manifestation of the geometry of spacetime.
Tidal forces
The equivalence principle, in its simple form, encounters a limitation when confronted with the gravitational fields produced by actual massive bodies.
The gravitational acceleration generated by a celestial body, for instance, is not a uniform vector field.
Instead, the acceleration vectors converge radially towards the body’s center of mass, and their magnitude diminishes with the square of the distance. Consequently, there exists no single coordinate transformation that can eliminate such a gravitational field globally.
However, the principle retains its validity in a local sense. A laboratory in free-fall within a gravitational field constitutes a local inertial frame.
For an observer inside this falling laboratory, the effects of gravity appear to vanish. Any object released within the laboratory will float alongside the observer, as both are accelerating identically towards the central body. Nevertheless, this cancellation is strictly local.
The impossibility of globally removing a real gravitational field can be understood by considering an extended object.
An object of non-infinitesimal size will experience a differential gravitational force across its structure.The portion of the object closer to the source of gravity is pulled more strongly than the portion farther away. This variation in the gravitational field creates internal stresses.
The object is stretched along the direction radial to the source and compressed in the transverse directions. These residual, non-transformable forces are known as tidal forces. Their presence is a definitive signature that gravity is more than a mere coordinate transformation.
The phenomenon of ocean tides is a direct consequence of such differential forces exerted by the Moon and Sun across the diameter of the Earth. It is therefore not strictly accurate to state that gravity is entirely equivalent to an accelerated reference frame on a global scale.
This does not imply a failure of Einstein’s reasoning. The equivalence principle is correctly understood as a local statement.
For a sufficiently small region of spacetime, that is, over small distances and for short durations, it is impossible for an observer to perform an experiment that distinguishes a uniform gravitational field from a uniformly accelerated reference frame.
This consideration brings forth a pivotal question: given an arbitrary force field, can one determine if a coordinate transformation exists that would make the field vanish everywhere? For the force field inside an elevator undergoing uniform acceleration, which was a constant vertical force, a non-linear coordinate transformation was sufficient to nullify it.
Consider another example: a carousel. An observer in this rotating reference frame experiences apparent forces, namely the centrifugal and Coriolis forces. The motion of a light beam or a free particle would appear quite complicated. This complex motion could be attributed to some effective gravitational field.
However, by performing the inverse coordinate transformation back to a stationary, inertial frame, this apparent field disappears, revealing its origin as a consequence of the non-inertial coordinates.
It is possible to generate a wide variety of fictitious gravitational fields through the selection of accelerating or rotating coordinate systems.
A related question is whether a field, known throughout a region, is a fictitious field resulting from a coordinate choice or if it represents a genuine gravitational field sourced by mass and energy.
In the Newtonian framework, the answer lies in calculating the tidal forces. If one were to place a sufficiently large crystalline structure in free-fall and measure any resulting stress or strain, its presence would confirm a real gravitational field.
The absence of such internal forces would indicate that the field is fictitious and can be eliminated by a suitable coordinate transformation.
Einstein realized that this physical question—whether a genuine gravitational field is present—is mathematically identical to the geometric question of whether a space is intrinsically curved or flat.
The tools to analyze such geometric properties were developed in the nineteenth century by Riemann. The presence of non-vanishing tidal forces is the physical manifestation of spacetime curvature.
This insight forms the bridge between the physics of gravitation and the mathematics of differential geometry, which provides the language for general relativity.