Uniform Acceleration in Special Relativity
What happens if you accelerate at a constant rate forever? In Newtonian physics, your velocity grows linearly: v = at. After enough time, you'd exceed the speed of light. But relativity forbids this. Instead, you follow a hyperbolic worldline that asymptotically approaches light speed.
The Worldline Equations
An observer experiencing constant proper acceleration a (measured in their instantaneous rest frame) follows these equations in inertial coordinates:
x(τ) = (1/a) cosh(aτ) − 1/a
t(τ) = (1/a) sinh(aτ)
where τ is proper time (time measured by the accelerating observer's clock). This traces out a hyperbola in spacetime. The velocity at any proper time is:
v(τ) = tanh(aτ)
As τ → ∞, tanh(aτ) → 1, so velocity approaches c but never exceeds it. You keep accelerating forever, but gains in velocity become smaller and smaller.
The Rindler Horizon
The hyperbola has an asymptote at x = −1/a. This is the Rindler horizon. Events to the left of this line can never catch up to the accelerating observer, no matter how long they wait. Light signals from these events never reach the observer.
Why Does a Horizon Form?
The accelerating observer's velocity keeps increasing, approaching c. Events behind the horizon would need to travel faster than light to catch up. Since nothing travels faster than c, there's a causal boundary — a horizon — separating events the observer can see from those they cannot.
Connection to Black Holes
This is remarkably similar to a black hole's event horizon! An observer hovering just outside a black hole must accelerate to avoid falling in. From their perspective, events inside the horizon are causally disconnected — just like the Rindler horizon. This analogy leads to deep connections in quantum field theory.
Rindler Coordinates
From the accelerating observer's perspective, spacetime looks different. They naturally use Rindler coordinates (ξ, η) related to inertial coordinates by:
t = ξ sinh(aη)
x = ξ cosh(aη)
In these coordinates, the observer is at rest (constant ξ), and η is their proper time scaled by acceleration. Surfaces of constant η are simultaneity surfaces for the accelerating observer. These are not horizontal lines — they're hyperbolas!
Proper Acceleration vs Coordinate Acceleration
Proper Acceleration
The acceleration you feel in your instantaneous rest frame. This is what an accelerometer measures. For a uniformly accelerating observer, this is constant. Units: m/s² or, in natural units, c²/c.
Coordinate Acceleration
The second derivative of position with respect to coordinate time: d²x/dt². For relativistic motion, this is not constant even when proper acceleration is! As velocity approaches c, coordinate acceleration drops toward zero.
This is why you can accelerate at 1g forever (proper acceleration) without ever exceeding c. Your coordinate acceleration keeps decreasing as you speed up.
Real-World Accelerations
Rocket Travel
A rocket accelerating at 1g (9.8 m/s² ≈ 10 m/s²) provides Earth-like gravity for passengers. After one year of proper time at 1g, you'd reach v ≈ 0.76c and have traveled about 0.5 light-years in the inertial frame. After 5 years, v ≈ 0.9999c.
Particle Accelerators
Electrons in circular accelerators experience enormous proper accelerations (10¹⁸ m/s² or more) due to centripetal force. They emit synchrotron radiation as a result. Their worldlines are helices in spacetime, constantly accelerating toward the circle's center.
Free Fall vs Acceleration
In general relativity, an object in free fall (like an astronaut in orbit) experiences zero proper acceleration. They're following a geodesic — the straightest possible path through curved spacetime. It's the ground beneath your feet that's accelerating you upward at 1g, preventing you from following a geodesic!
Things to Try
Increase Proper Time: Watch how the observer's worldline curves as they accelerate. Notice how velocity increases rapidly at first, then asymptotically approaches c. The coordinate distance traveled keeps growing, but velocity gains slow.
Vary Acceleration: Higher acceleration means a tighter hyperbola and a closer horizon. At a = 10, the horizon is only 0.1 light-years behind you. Lower acceleration (a = 0.1) creates a very gentle curve with a distant horizon.
Observe the Horizon: Events behind the Rindler horizon can never send signals to the accelerating observer. Place a light source behind the horizon — its light never catches up. This is a true causal boundary.
Simultaneity Surfaces: Enable simultaneity lines to see surfaces of constant proper time for the accelerating observer. These are hyperbolas! Events simultaneous in the accelerating frame are not simultaneous in the inertial frame.
Long-Term Acceleration: Set proper time to 5 and acceleration to 1. The observer has been accelerating for 5 years (their time) and is now at v ≈ 0.9999c. In the inertial frame, much more coordinate time has passed due to time dilation.
Unruh Effect: Temperature from Acceleration
Here's something bizarre: an accelerating observer in empty space sees a thermal bath of particles at temperature:
T = (ℏa)/(2πckʙ)
where ℏ is Planck's constant and kʙ is Boltzmann's constant. This is the Unruh effect. An inertial observer sees nothing (the vacuum), but an accelerating observer sees thermal radiation!
For a = 1g, the Unruh temperature is absurdly tiny: T ≈ 4 × 10⁻²⁰ K. But for extreme accelerations (a ~ 10²⁰ m/s²), it becomes measurable. This effect is deeply connected to Hawking radiation from black holes.
Connection to General Relativity
Einstein's equivalence principle states that acceleration is locally indistinguishable from gravity. An accelerating rocket feels exactly like sitting on a planet's surface. This means:
- Time dilation from acceleration = time dilation from gravity
- Rindler horizon = event horizon (locally)
- Unruh radiation = Hawking radiation (same mechanism)
Studying Rindler spacetime (flat spacetime with acceleration) teaches us about curved spacetime and black holes. The mathematics is simpler, but the physics is deeply related.
Philosophical Implications
The Rindler horizon shows that horizons aren't just a feature of black holes or cosmology — they emerge from pure kinematics. If you accelerate forever, there will always be events you can never see. The observable universe for an accelerating observer has boundaries.
This suggests that horizons are fundamental to physics. In quantum gravity, some theories propose that spacetime itself emerges from information accessible to observers. Horizons define what information is accessible, making them central to the structure of reality.
Why This Matters
Uniform acceleration is the simplest non-inertial motion in special relativity. It reveals:
- How velocity asymptotically approaches c under constant force
- That horizons can exist in flat spacetime
- The deep connection between acceleration and gravity
- How different coordinate systems describe the same physics
Understanding Rindler spacetime is essential for studying general relativity, black hole physics, and quantum field theory in curved spacetime. It's the bridge between special relativity and Einstein's theory of gravity.