The Wait Equation

Short description: Thought experiment about timing interstellar missions

The Wait Equation, also known as the Waiting Problem or the Relativistic Race Paradox, is a thought experiment in astronautics and theoretical futurology that determines the optimal time for a civilization to launch a mission to another star system.

The concept is an optimization calculation used to find the minimum total time required to reach a distant destination, given the assumption of continuous, exponential technological change and growth in propulsion systems.

Origin and development

The core concept that later ships, using better technology, could overtake earlier, slower ships was inspired by the works of science fiction author and physicist Robert L. Forward in the late 20th century. The idea was formally developed and the term "Wait Equation" was coined by astrophysicist **Andrew Kennedy** in the early 2000s, who created a mathematical model to calculate the optimal delay for a starship launch.

The premise is that launching a generation ship today with slow, sub-light speed technology risks having that mission rendered obsolete when a later generation launches a much faster, technologically superior ship after a period of waiting and R&D.

Equation and calculation

The Wait Equation seeks to find the period of time $W$ (the "wait time") that minimizes the total time to destination ($T_{total}$), which is the sum of the time spent waiting on Earth plus the time spent traveling ($T_{travel}$).

$T_{total}=W+T_{travel}$

The total travel time is determined by the distance $D$ divided by the velocity $v$. Assuming the velocity increases exponentially based on technological growth rate $r$ and the wait time $W$:

$T_{total}=W+\frac{D}{v_0\cdot (1+r{)}^{W/k}}$

Where:

• $T_{total}$ is the total time from now until arrival.
• $W$ is the time spent waiting on the home planet.
• $D$ is the distance to the star system.
• $v_0$ is the initial maximum velocity available today.
• $r$ is the average annual growth rate in propulsion efficiency.
• $k$ is a constant related to the time it takes for technological improvement to double the available velocity.

The equation demonstrates that for certain assumed high rates of technological growth, delaying the launch by $W$ years results in a much greater reduction in $T_{travel}$, causing the total arrival time $T_{total}$ to be minimized at a specific optimal wait time.

Notability and implications

The Wait Equation is notable for several reasons, primarily as a practical challenge to the immediate viability of slow interstellar travel.

• The Incentive Trap: As long as a civilization has a reasonable expectation of continuous technological growth, no one has an incentive to be the first to launch, as they risk making a multi-generational sacrifice only to be surpassed by later, faster voyages. Kennedy argues that this acts as an "incentive trap" that could delay serious interstellar colonization efforts.
• Resource Allocation: The equation provides a quantifiable argument for focusing resources on fundamental breakthrough physics and propulsion technology rather than incremental engineering improvements on slow ships. The logic dictates that the "best path to the stars is through the laboratory, not the launchpad."
• Analogy for Technological Acceleration: In the 21st century, the concept has been widely applied as a metaphor for the rapid acceleration of technology, particularly in the field of Artificial Intelligence (AI). In this analogy, the "slow ship" represents traditional human academic or professional effort, while the "fast ship" represents a new AI system that, developed after a short "wait," automates or solves problems at hyper-speed, rendering the slow human effort obsolete.

See also

• Fermi paradox
• Great Filter
• Bracewell probe