Fighter-jet pilots can survive an acceleration ten times larger than the gravitational acceleration on the surface of Earth, 10 g. Electronic components of artillery shells can be hardened to withstand up to a hundred thousand g
The surface gravity of a white dwarf, the future remnant of the Sun, is also a hundred thousand g, about 30 times larger than the acceleration of a baseball struck by a bat. The surface gravity of the most compact star with a hard surface, a neutron star — typically 12 kilometers in radius, is of order a hundred billion g, about 20,000 times larger than that of a jellyfish stinger.
The accelerated expansion of the Universe is slow, reaching across our cosmic horizon to a value equal to the speed of light over the age of the Universe. Remarkably, by some cosmic coincidence, this happens to also be the acceleration of the Sun around the center of the Milky-Way galaxy. But what is the largest acceleration in the Universe?
The largest accelerations can be gauged in terms of the time needed to gain the largest possible speed, namely the speed of light. This acceleration time can in turn be expressed as an acceleration length over the speed of light. And so altogether, the largest accelerations in nature can be expressed as the speed of light squared over the shortest acceleration length possible.
What is the shortest acceleration length possible? In quantum mechanics, every particle of mass m is characterized by a wave function of a spatial extent larger than the reduced Compton wavelength, ℏ/mc, where ℏ is the reduced Planck constant and c is the speed of light. This implies that the highest acceleration possible for an elementary particle is the speed of light squared divided by the reduced Compton wavelength of the particle.
For charged particles, like electrons or positrons, the corresponding maximum acceleration is 25 octillion (=2.5 times 10 to the power of 28) g. This acceleration can be achieved by a strong enough electric field. In fact, an electric field of that strength creates electron-positron pairs out of the vacuum because it amounts to an electric potential drop of twice the electron mass across the reduced Compton wavelength of the electron. A field stronger than this value breaks down into electron-positron pairs. In 1951 this effect was theoretically calculated by Harvard physicist and Nobel laureate Julian Schwinger. Electric fields approach this critical value near rapidly spinning neutron stars, where pair-production leads to pulsing radio emission, giving rise to pulsars. The fastest spin of known pulsars is 716 rotations per second for the pulsar PSR J1748–2446ad.
Since the Compton wavelength goes down with increasing particle mass, particles more massive than the electron could potentially reach higher accelerations. For example, the proton can exceed the electron maximum acceleration by a factor of 1836.15, the ratio between the proton and electron masses.
If dark matter is made of particles that are even more massive than the proton, then their maximum acceleration can be even larger. However, this would only be possible if these unknown particles respond to some unknown dark force that can accelerate them by that much.
One acceleration mechanism that applies to all particles is gravity. Gravitational acceleration is strongest near the event horizon of a black hole. But to exceed the maximum acceleration of a proton, one needs the event horizon to be smaller than the Compton wavelength of the proton. This occurs for primordial black holes with a mass lower than 0.6 billion tons. However, smaller black holes evaporate by Hawking radiation over a timescale shorter than the age of the Universe. As a result, they are not expected to be found in the present-day Universe.
The shortest possible wavelength for a particle is the Planck length, 33 orders of magnitude smaller than a centimeter. The associated maximum acceleration attainable for this length is 0.6 sextillion nonillions (=6 times 10 to the power of 50) g.
Composite objects made of many particles cannot exceed this Planck limit because the signal that they are accelerated must cross them in order for them to move as a single unit, and the shortest crossing time is achieved at the speed of light. The smallest size for an object of a given mass is the horizon size of a black hole with that mass. The smallest black hole possible is the one for which the Compton wavelength equals the horizon size. This happens to be a black hole with the Planck mass, 21.76 micrograms. Hence, within the known theories of quantum mechanics and gravity, the acceleration limit is the speed of light squared divided by the Planck length. The Planck acceleration is the absolute limiting acceleration based on the known laws of physics.
But let us get back to Earth. Surprisingly, the modest 1g acceleration to which the human body is accustomed, fits well the goal of interstellar travel. By accelerating at 1g for one year, a traveler can reach the speed of light. This boost would allow the interstellar traveler to reach up to a hundred thousand stars during a human lifetime. However, if the spacecraft engine continues the uniform 1g acceleration for longer than a year, the travelers could reach cosmological distances over decades because of time dilation in their frame.
To find out if any cosmic traveler is moving near the speed of light, we can survey the sky with our telescopes and gravitational-wave observatories. In a recently published paper, I showed that the existing gravitational-wave observatory LIGO can detect the gravitational signal of a massive stealth spacecraft that moves at the speed of light in the vicinity of Earth.
ABOUT THE AUTHOR
Avi Loeb is the head of the Galileo Project, founding director of Harvard University’s — Black Hole Initiative, director of the Institute for Theory and Computation at the Harvard-Smithsonian Center for Astrophysics, and the former chair of the astronomy department at Harvard University (2011–2020). He is a former member of the President’s Council of Advisors on Science and Technology and a former chair of the Board on Physics and Astronomy of the National Academies. He is the bestselling author of “Extraterrestrial: The First Sign of Intelligent Life Beyond Earth” and a co-author of the textbook “Life in the Cosmos”, both published in 2021. His new book, titled “Interstellar”, was published in August 2023.
