Monday, December 13, 2010

What Everybody Knows about Einstein

Back in 1905 a young clerk at the Swiss Patent Office thought deeply about the behavior of bodies traveling close to the speed of light.  He showed mathematically that the observed properties of such bodies could be powerfully affected by the relative motion of observer and observed, with the observations deviating ever more profoundly from low-speed observations as the relative velocity approached the speed of light.

Specifically, observations of lengths, masses, and time duration could all be dramatically affected by relative motion. It follows that different observers of the same event would observe very different masses, times, and distances. But Einstein did not stop there. He provided a set of transformation equations by which different observers could correct for these effects and “translate” their observations into a form comprehensible to others. Having provided a precise quantitative means for observers to understand each other’s observations, Einstein summarized the situation by propounding the Relativity Principle: “The laws of the Universe are the same for all observers”.

Everybody known that Einstein said that “everything is relative.”  He most certainly did not.  That would provide a convenient excuse for not trying to understand anything; for maintaining that all perspectives are equally valueless, and the Universe is fundamentally incomprehensible. Such sloppy thinking permits newspaper readers to conclude that there are no absolutes and that this generalization extends to morality as well as physics.  But this understanding of the Relativity Principle is precisely backwards, and its application to human behavior is almost hilariously inept.  

Everybody also knows that “Einstein showed that Newton was wrong”.  This is another example of why people should not use newspapers as science text books.  The classical example of how Newton was wrong is mass dilation.  Einstein showed that observations of the mass of an object traveling close to the speed of light show a dramatically increased apparent mass.  Experiments bear this out: the faster a body is moving, the more energy is required to increase its speed by a given amount.  The apparent inertial mass of a body becomes infinite at the speed of light, which means that infinite energy would be required to accelerate a body to the speed of light.  Now Newton surely never said or suspected any such thing.  Let’s go back and look at what Newton said about accelerating bodies.

Everybody knows that Newton said “F = ma”; the force required to accelerate a body at rate a is equal to the inertial mass of the body.  Curiously, however, Newton did not say that (thousands of textbooks notwithstanding).  Newton actually stated his Second Law in the following terms: “force is the rate of change of momentum”.  In calculus (co-invented by Newton and Leibnitz) the statement is F = d(mv)/dt, which mathematically implies F = v(dm/dt) + (dv/dt).  Generations of introductory physics texts have said in effect, either explicitly or implicitly, “if the mass is constant, then dm/dt is zero, so the term v(dm/dt) is also zero.  Then F = m(dv/dt), and the rate of change of v, (dv/dt), is obviously the acceleration, a.  Thus F = ma, and the acceleration of a body of mass m acted upon by a force F is a = F/m.

But Newton did not go there.  In his theory, a body with mass m would obey the equation F = v(dm/dt) + ma, so the acceleration of the body under the action of force F would be a = [F - v(dm/dt)]/m.  This equation holds as well in relativistic (Einsteinian) physics as in classical (Newtonian) physics!  What Einstein provided was the concept that the energy imparted to a body in the process of acceleration to high speeds adds equivalent mass to the particle: the more energy imparted to the particle, the greater its effective mass, the greater its inertia, and the less its acceleration under the action of a constant force.  In other words, the change in mass of a particle caused by a change in its energy is the energy change divided by the square of the speed of light.  The newspapers report this as E = mc2

Relativity is based not only on the “rest” properties of an object (mass, dimensions, and time intervals measured by an observer at rest relative to the object) but also on the equations used to translate the observations of different observers and the universal, absolute principle that not only are there laws, they are the same for all observers.

The next time someone tells you that everything is relative or that the Universe has no absolutes, fell free to laugh.



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Anonymous said...

Another frequently-repeated error (which, shamefully, is even found in many school textbooks) is the statement that sub-orbital trajectories are parabolas, when such trajectories are actually closed, elliptical orbits that happen to intersect the surface of a planet. This is a particularly egregious error given that the parabola, the trajectory of an object that has *exactly* enough velocity to escape from a celestial body, is an "ideal" path that, like a perfectly circular orbit, does not occur in the real universe.