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15 August 2003

Why Can't We Go Faster?

by George Hrabovsky, President of MAST

News from MAST

The news from MAST is short this time.  I am preparing a CD-ROM version of The Mind of a Theorist for purchase from the MAST website.  This version will include the MathReader software and all of the first volume of notebooks.  I will be adding appendices on how to use Mathematica.  The release date for this volume is still a bit murky, I will let you all know.

How Does Mass Behave in Relativity?

So far we have shown that the most fundamental quantities of kinematics such as time and length depend on the state of the observer.  Does this work for dynamics too?  The most fundamental quantity in dynamics is mass.  Assume that a particle has some mass m and is traveling with respect to an inertial frame at some velocity v, then it seems reasonable to write an expression for mass that tells us that the mass too is observer-dependent,

m = m (v) .

The problem is to find what this functional dependence is.

Let us see what happens when this particle strikes another particle so that they stick together, such a collision is termed inelastic.  Let us state that this struck particle is in the rest frame of observer , the other particle is moving as stated above.  After the collision the combined objects move with the new velocity V.  By our function above we note that the masses of the two particles are m(0) and m (v) respectively.  In our case we decide that we can rewrite the non-moving particles mass as m_0 and, since it is in the rest frame, we will call it the rest mass.  We will call the mass following the collision M (V).

Let us further assume that the second observer,  ', is in the frame of the center of mass of the system of particles (see, "Getting to the Center of Things" for a review of the idea of center of mass.)  In this frame the result of the collision will be at rest with the new rest mass M_0.  By implication we can see that  ' must be moving at velocity V with respect to .

We assume that no mass is lost in the collision and that the idea of conservation of momentum holds (see, "The More Things Change, The More Some Things Stay The Same," for a review of conservation of momentum.)  In the frame of  we see that we have,

FormBox[RowBox[{m (v) + m_0, , =, , RowBox[{M (V), Cell[       ...             (1)]}]}], TraditionalForm]

and, if we multiply the masses by their respective velocities we get

m (v) v = M (V) V .

We can rewrite this

m (v) v = [m (v) + m_0] V

or

m (v) v = m (v) V + m_0V .

We can rewrite this,

m (v) v - m (v) V = m_0V .

We can factor this,

m (v) (v - V) = m_0V

and then rewrite it,

m (v) = m_0V/(v - V) .

Addition of Velocities

We now have the problem of combining two velocities.  While it is tempting to just add them, we have been seeing a lot of things in relativity that are not intuitive.  Making assumptions of normality is dangerous.  Recall the velocity definition,

V ' = (Δ x ')/(Δ t ') .

From the Lorentz transformations this becomes.

V ' = (γ (Δ x - v Δ t))/(γ (Δ t - v Δ x))

      = ( Δ x - v Δ t)/( Δ t - v Δ x)

      = ( Δ x/Δ t - v)/( 1 - v Δ x/Δ t)

      = ( V - v)/( 1 - v V) .

This is called the Composition of Velocities Law.

Relativistic Mass Revisited

The particle in motion initially has velocity V compared to the rest frame, which in turn has the same velocity relative to .  By the law of composition of velocities we have

( V - v)/( 1 - v V) = ( V - (-V))/( 1 - (-V) V) = ( 2 V)/( 1 + V^2) .

In non-natural units this becomes

( 2 V)/( 1 + V^2/c^2)

This must be equivalent to the initial velocity of the particle,

v = ( 2 V)/( 1 + V^2/c^2)

We can solve this for V, resulting in the quadratic equation

FormBox[RowBox[{V^2 - ((2 c^2)/v) V + c^2, , =, , 0.}], TraditionalForm]

whose solution is,

V = (c^2 + (c^4 - c^2 v^2)^(1/2))/v

       = c^2/v ± (c^4/v^2 - c^2)^(1/2)

       = c^2/v ± c^2/v (1 - v^2/c^2)^(1/2) .

Substituting this result into

m (v) = m_0V/(v - V)

we get

m (v) = m_0 (c^2/v + (c^2 (1 - v^2/c^2)^(1/2))/v)/(v - c^2/v + (c^2 (1 - v^2/c^2)^(1/2))/v) .

Wow! Is that ever ugly looking.  Fortunately, this boils down to,

m (v) = γ m_0 .

Thus we see that mass seems to increase with velocity.  Since mass increases with velocity the quantity of kinetic energy required to move will increase.  Also note that as v gets closer to the speed of light γ gets closer to infinity.  We will discuss more of the ramifications of this later.

Answer to the Theory Challenge from Last Time

The task was to find the work is done by an oscillator in the steady state?  The work is given by

W = Overscript[x, ·] F(t) .

Recall that,

F(t) =   A_0 cos (ω t + θ_0)

and

x = A ^(-γ t) cos(ω_1 t + θ) + Re[A_0 ]/m [(ω_0^2 - ω ^2)^2   + 4γ^2 ω^2]^1/2  sin(ω t + θ_0 + β) .

so

Overscript[x, ·] = -A ^(-t γ) γ cos(ω_1 t + θ) + (ω cos(` ... 2 + (-ω^2 + ω_0^2)^2]^(1/2) - A ^(-t γ) sin(ω_1 t + θ) ω_1 .

This can be simplified,

Overscript[x, ·] = (ω cos(ω t + θ_0 + β) Re[A_0])/m[4 γ^2 ω^2 + ... ) - A ^(-t γ) (γ cos(ω_1 t + θ) + sin(ω_1 t + θ) ω_1) .

The work done is then

W = A_0 cos (ω t + θ_0) [ (ω cos(ω t + θ_0 + β) Re[A_0])/m[4 γ^2 ω^2 + (ω^2 - ω_0^2)^2]^(1/2) -

        A ^(-t γ) (γ cos(ω_1 t + θ) + sin(ω_1 t + θ) ω_1)] .

Note that the last term rapidly becomes small enough to ignore over time.  When this vanishes, the work for the steady state part is left,

W = A_0 cos (ω t + θ_0) (ω cos(ω t + θ_0 + β) Re[A_0])/m[4 γ^2 ω^2 + (ω^2 - ω_0^2)^2]^(1/2) .

Theory Challenge

What is the power of the oscillator?

Books That I Like

Bernard F. Schutz (1990), A First Course in General Relativity, Cambridge University Press. This has a nice introduction to spacetime diagrams and derives how to view other coordinate systems.

Ray D'Inverno (1992), Introducing Einstein's Relativity, Oxford University Press. This is my favorite introduction to relativity.

Online Resources

4-Vectors:

For a brief description go to:

http://www-physics.mps.ohio-state.edu/~cleo/hep/explain/4vector_1.html

For a slightly more advanced approach,

http://www.pact.cpes.sussex.ac.uk/users/markh/RQF1/node12.html

Forced Oscillators

Here is a fairly elementary site that presents the topic in general terms.

http://www.pinkmonkey.com/studyguides/subjects/physics/chap10/p1010601.asp

Here is a more challenging site that gets similar results to those above.

http://colos1.fri.uni-lj.si/~colos/COLOS/TUTORIALS/JAVA/JAVAXYZET/RESONANCE/HTML/resonance_6.html

Created by Mathematica  (August 14, 2003)