Relativity and Planet of the Apes

“Seen from out here, everything seems different, time bends, space is boundless, it squashes a man’s ego.” –  Charlton Heston in The Planet of the Apes on the relativistic effects of traveling near the speed of light.

The Statue of Liberty just celebrated its 130th birthday which reminded me of the famous ending of the original Planet of the Apes.  For me, the beginning of this movie is important as it was the first time I had encountered the concept of relativity and time travel.  That is, time will move more slowly for a person in motion than for a person who is stationary.  This effect is not noticeable with the slow velocities in which we travel on Earth but becomes more pronounced when moving towards the speed of light.  And give Planet of the Apes credit, it gets it right, unlike say Star Trek, which often takes a cavalier attitude towards relativity for dramatic purposes.  The video below is the beginning two minutes where this plot device is introduced.

One caveat here, even during the height of the Mad Men era, NASA did not allow smoking during its missions.  The scientist mentioned, Dr. Hasslien, is a fictitious character.  The chronometer puts the ship year at 1972 but the Earth year at 2673.  By the time the ship lands, it is the year 3978.

So how does this premise work?  We can start by looking at Einstein’s time dilation equation:

Δt’ = Δt/[1 – (v2/c2)]1/2  where:

Δt’ = time elapsed on Earth

Δt = time elapsed on spacecraft

v = velocity of spacecraft

The exponent of 1/2 is another way of saying square root.

c = speed of light (3 x 108 m/s or 186,282 miles per second)

When an object is stationary (v = 0) the denominator on the right side equals one.  Thus, Δt’ = Δt and both clocks run at the same rate.  As v approaches c, the term v2/c2 approaches 1.  This increases the value of the right side of the equation meaning Δt’ must increase to keep both sides of the equation equal.  Lets take a look at a couple of examples.

The velocity of the International Space Station is about 5 miles per second or 8000 m/s.  What is the time dilation effect of an astronaut who spends a year aboard the station?

Δt = one year or 3.15 x 107 seconds

v = 8000 m/s

Plugging into the equation gives:

Δt’ = 3.15 x 107 s/[1 – (8000 m/s)2/(3 x 108 m/s)2]1/2

Δt’ = 3.15 x 107 s/[1 -(6.4 x 107 m2/s2/9.0 x 1016 m2/s2)]1/2

Before the final calculation, a couple things to note.  You have to standardize your dimensions before calculating.  In physics, this usually means converting to meters/kilograms/seconds.  Not doing this is a common mistake for students taking their first physics course.  Also, the term m2/s2 cancels out leaving us with only seconds in the answer.  Since we are measuring time, checking dimensions will make sure you are on the right track. So, the answer is:

Δt’ = 3.15 x 107 s/[1 -(7.11 x 10-10)]1/2

Δt’ = 3.15 x 107 s (0.99999999964)

Δt’ = 31499999.99 s

So on Earth, our clocks advanced 31,500,000 seconds and the astronauts in orbit clocks advanced 31,499,999.99 seconds, so the ISS astronaut would have aged about 1/100 of a second less than us on Earth.*  What would happen if you were to spend a year traveling at  99% the speed of light?  Here, we can use fraction of light speed in the equation as the dimensions will drop out.

Δt’ = 3.15 x 107 s/[1 – (0.98c/1c)]1/2  0.98 being 0.99 squared.

Δt’ = 3.15 x 107 s/(0.02)1/2

Δt’ = 3.15 x 107 s/(0.141)

Δt’= 223,404,255 s or 7.1 years

If we up the speed to 99.9% of light speed, Δt’ becomes 22.3 years.  To get the time dilation effect seen in Planet of the Apes you would need to travel about 99.99999% of light speed.  The graph below shows the time dilation effect with changing velocity.

Credit: Wiki Commons
Credit: Wiki Commons

You’ll note the time dilation effect does not show up significantly until you reach 40% of light speed or about 75,000 miles per second.  That speed would get you to the Moon in 3 seconds.  The effect has an upper bound at the speed of light.  That is, the time dilation effect approaches infinity as velocity nears light speed.  In fact, once you hit the speed of light, your clock would stand still.  And there’s no going back.  The time travel possibility is a one way ticket forward as going faster than light speed is required to move backwards in time.  In Einstein’s universe, nothing can travel faster than light speed.  The reason for this is mass increases when velocity increases.

Newton’s second law states that force is equal to mass times acceleration.  The assumption here is that mass is constant and thus, all the force results in accelerating an object.  Einstein discovered that as an object approaches light speed, mass is not constant and approaches infinity.  The equation to determine mass with velocity is as follows:

m = m0/[(1 – v2/c2)]1/2

m0 = rest mass

m = mass in motion

When velocity is 0, m = m0.  To apply this to the Planet of the Apes scenario, lets assume the mass of the space vehicle is the same as the Apollo command/service module at 15,000 kg (33,000 lbs).  If we accelerate to 99.99999% of light speed, its mass would increase to 33.5 million kg (74,000,000 lbs) or about 12 Saturn V rockets.  At this point, more force gets decreasing returns in velocity as the spacecraft’s mass increases and becomes more difficult to push.

The term (1 – v2/c2)1/2  is referred to as the Lorentz transformation and is frequently seen in special relativity equations.  For shorthand, is is often symbolized by γ.  Besides time and mass, length is also impacted by velocity and contracts as an object approaches light speed.  The Hyperphysics website has some nifty relativity calculators you can check out here.

Our first attempts to reach another star will not be in large starships such as the U.S.S. Enterprise of Star Trek fame.  More than likely, it will be in a fleet of tiny spacecraft such as proposed by Stephen Hawking for Operation Starshot.  Using nanotechnology, the goal is to send thousands of 20 gram (about 0.7 oz.) probes to our nearest interstellar neighbor Alpha Centauri.  Light sail technology would propel these vessels to 20% of light speed.  At this rate, the mass of each probe would only increase from 20 to 20.4 grams.  Even if velocity reached 80% of light speed, the mass increase would only be to a manageable 32 grams.  Having thousands of smaller probes rather than one large craft increases the odds that the mission reaches its final destination even if some get damaged along the way.

To sum it all up, the faster you move through space, the slower you move through time.  Also, motion brings about an increase in mass.  Both these effects do not become pronounced until you reach 40% light speed, which does not happen to us here on Earth.  Time stands still at the speed of light and mass approaches infinity as you close in on light speed.  This makes human travel to the stars very problematic.  Of course, in The Planet of the Apes, the crew basically made a round trip to Earth.  Charlton Heston discovers that when happening across the ruins of Lady Liberty.

Never did understand why all those apes speaking perfect English did not clue him in to that beforehand.

*If we were to delve into general relativity, gravity slows clocks the same as velocity does as seen in Interstellar.  This means being on a planet surface with greater gravity slows your clock compared to someone in orbit.  This offsets the velocity time dilation for astronauts in orbit.  Factoring the two, astronauts age about a millionth of a second less than us here on Earth.

**Photo atop post is the chronometer on Heston’s spacecraft.  Credit: 20 Century Fox.

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