22.3. Simultaneity http://www.ck12.org
There are several arguments demonstrating that special theory of relativity does not create a paradox and, indeed,
that it is the twin on the ship who ends up younger. It’s important to recall that the special relativity equations above
are valid only for the inertial frames of reference, or frames that do not accelerate. In order for the traveling twin
to return to the Earth, she must turn around. Any change in direction indicates a change in velocity and, therefore,
acceleration.
Even though the situation has an interval of acceleration, one can still use special relativity to correctly determine the
result. The inertial frame returning may not be the same inertial frame when leaving, but they are both still inertial
frames. And if the traveling twin never returns, thus introducing no acceleration, the paradox becomes mute since
there is then no way for the twins to compare ages.
Of course, nothing prevents one from arguing that it was the Earth that underwent the acceleration, and the spaceship
that remained motionless. But how would the Earth do this? The only way the Earth could move back and forth is
to suppose the entire universe must be moving back and forth. Even with this assumption, it can be shown using
Einstein’s general theory of relativity, which deals with accelerating reference frames, that the results of special
relativity are still correct. The situation between the Earth and the rocket ship is not symmetrical.
Only the traveling twin has experienced acceleration –therefore, the twins’ situations are not equivalent, and the
home-bound twin will, in fact, be older. Hey, you can’t expect to find eternal youth by staying home, right?
http://demonstrations.wolfram.com/LorentzTransformationForTwinParadox/
Mass-Energy
Using special relativity, force and momentum must also be altered with the multiplicative factor√^1
1 −vc^22
.
The relativistic equations for force and momentum are:
F=ma→F=
ma
√
1 −v
2
c^2
p=mv→p=
mv
√
1 −v
2
c^2Using special relativity, Einstein was also able to show that mass and energy were not independent of each other, but
rather, equivalent. In fact, energy could be converted to mass and vice versa.
We state without proof that Einstein derived the relationship between energy and mass asE= mc
√^2
1 −vc^22
.
The termmc^2 is called the rest energy of the object. Einstein reasoned that since mass and energy were equivalent,
a resting object of massmwould have an equivalent energymc^2. Thus, even when an object has no kinetic energy,
it still has energy by virtue of the fact it has mass.
The total energyEof the object can be rewritten as
E=mc^2 +KE
The total energyEof an object is equal to the sum of the rest energy of the objectmc^2 , and the kinetic energy of the
objectKE. Notice that if theKEof the object is zero, the total energy of the object reduces toE=mc^2 , Einstein’s
iconic equation.
We can also show that the relativistic equationE=mc^2 +KEreduces to the Newtonian equation for the kinetic
energy whenv c. We show this result in the following example.