Thursday, April 24, 2014

The Famous Twin Paradox

Maybe you have heard about the fantastic story of twin brothers who part ways on their twentieth birthday.  Let's call them Billy and Timmy.  Billy stays on Earth, while Timmy travels at 99.9% the speed of light on his fancy space ship.  Timmy returns to Earth on Billy's fortieth birthday after having explored the universe.  Timmy is visibly younger than Billy.  In fact, he has only been travelling for 326 days as far as he is concerned.  He is yet to turn 21 - he is not legal to consume alcohol in the United States though his ID says he is middle-aged.

Why the time discrepancy?  It has everything to do with speed.  Special relativity, as theorized by Einstein in 1905 and confirmed countless times by modern experiments, says that the faster we move, the slower time passes for us.  This is not science fiction; this is science.

The time ratio, also known as the Lorenz factor, which gives a ratio of the passage of time between stationary and moving observers, is given as follows...

 
Here, v is the speed of the moving observer, while c is the speed of light.  If we compute the Lorenz factor between Billy and Timmy, it is 22.37.  Meaning that although twenty years passed for Billy, only [1 / 22.37] of this amount passes for Timmy, which results in about 326 days.  For more insight into the phenomenon of time dilation, try this link.

While it is fun to be swept away by the fascinating twin paradox, let us stop for a moment and investigate the plausibility of the scenario as well as one factor that is conveniently overlooked in its conception.

The first point to address is, "Can a space ship that travels at 99.9% the speed of light (299,700 km/s) be built?"  Conventional rockets could never do this of course.  But, how about decades, or even centuries from now - could it be possible then?  The trouble is the amount of energy that is required to bring an object from rest to this speed.  This amount is given by: Work = (ɣ-1)mc2

Here, m is the mass of the spacecraft and all of its contents.  In Timmy's case, let us say that this amount is 10,000 kg.  Then, the work that must be input by the 'super engine' is 1.92E22 J or about 20,000 billion billion Joules (even more if we note that no engine is 100% efficient in converting its energy into work).  To put this into perspective, all of mankind used about 2.7% of this amount of energy during the year 2008.  In fact, if we added up the total amount of energy used by mankind in all of time (aka since the industrial revolution) we would get in the ballpark of what Timmy requires to get his spacecraft moving near the speed of light once.

This makes the adventure seem implausible, at least for the foreseeable future.  Who knows, maybe this quantity of energy will become accessible in the distant future... And, once the ship moves at the high speed, we could recover some or even most of the energy as it returns to rest.

It seems to me though, that even if we had a spacecraft that were up to the task and an energy supply that met the demand, there remains a problem with the twin paradox.  It is a simple kinematic problem combined with the limitations of human physiology.

It takes time to accelerate an object from rest to 299,700 km/s.  The slower we accelerate, the more time it takes.  A person is comfortable at 1-g (10 m/s/s) but uncomfortable much beyond it.  We'd feel sick at 3-g and surely unconscious at 5-g for a long duration (poor Timmy).  Let us say that Timmy's spacecraft accelerates at a rate of 20 m/s/s, then it will take 173 days to reach the 0.999c cruise speed.  It would take the same amount of time to decelerate comfortably to rest at the end of the trip.

In this case, for about one year of the travel, Timmy is not at the super high speed where time dilation effects are significant.  If we are realistic about Timmy's acceleration, we must add about one year to his tour of the universe.  That means that he would more realistically be about 21 years and 326 days upon his return, and could in fact legally have a beer to celebrate his return with his paradoxically older twin brother.

2 comments:

Ed Wolofsky said...

Why couldn't the twins be Jay and Rob?...just asking!

The Engineer said...

(Hey Ed) By Einstein's first postulate associated with special relativity, the laws of physics apply equally to everyone in the universe, regardless of their reference frame. So ya, it could also be Jay and Rob ;)