Relativity and Angular Momentum

The idea that orbital velocity is affected by relativity is central to the theory presented in Sampling the Hydrogen Atom, so it is perhaps worthwhile examining this idea in a little more detail.  Before doing so however it is necessary to restate that the use of Special Relativity in dealing with objects which have constant orbital velocity is entirely appropriate, this despite the fact that such objects are subject to acceleration.  The velocity of an object which is in orbit can be considered as having two components, a tangential component and a radial component.  For constant orbital velocity, the tangential component is itself constant and therefore can be dealt with using Special Relativity which affects the time and distance measured along the orbital path.  Direct evidence to support this comes in the form of the Muon ring experiment described earlier.

Such an orbiting object is subject to constant acceleration towards the orbital centre and it is this acceleration which in effect maintains the circular path.  Conventional wisdom has it that this centripetal acceleration is not affected by relativity, since it acts in a direction which is normal to the velocity of the object.  Here it is argued that this cannot be the case since the distances involved in calculating centripetal acceleration derive directly from the distances travelled around the orbital path and that these distances are themselves affected by relativity.  It can then be shown that this is equivalent to substituting Relativistic Velocity in place of Actual Velocity in the standard formula for calculating centripetal acceleration.

Einstein showed that objects which are traveling at close to light speed are affected in three ways, time in the domain of the moving observer advances at a slower rate than it does for a stationary observer, distance for the moving object is foreshortened in the direction of travel relative to that same distance as measured by the stationary observer.  The mass of a moving object appears increased as far as the stationary observer is concerned.  All three effects occur to the same extent by the factor Gamma (γ).  Gamma is named after the Dutch physicist Hendrik Antoon Lorentz (1853 – 1928).  Gamma is given by the formula

  Equation 1

Examination of the effect of relativity on an object moving at close to the speed of light however reveals that both time and distance are scaled by a factor 1/γ and so from Equation 1

  Equation 2

It can be seen that this is the equation of a circle, more specifically a quadrant of a unit circle, since v is constrained to lie between 0 and c as shown in Figure 1.

Figure 1
(Click on image to enlarge)

If the object under consideration is in circular orbit, then this quadrant can be superimposed on the orbital path to form a hemisphere.  Objects orbiting at non-relativistic speeds see the path length around the orbit as being equal in length to the equator, while objects orbiting at higher speeds follow a path length described by a line of latitude on the hemisphere.  An object orbiting at the theoretical maximum speed of light would then be pirouetting at the pole.  We can consider the length of the orbital path as being represented by the line of latitude formed by a slicing plane which cuts through the hemisphere parallel to the equatorial plane.  In Figure 2 this is at approximately 15% of the speed of light c and so the orbital path length is just a little less than the equatorial path length, around 99%.

 Figure 2
(Click on image to enlarge)

In Figure 3 the orbital velocity is approximately 80% of the speed of light and so the orbital path length as seen by the moving object is approximately 60% that for an object moving at non-relativistic speed.

Figure 3
(Click on image to enlarge)

In Figure 4 the orbital velocity is around 98% of the speed of light and the corresponding orbital path length is approximately 20% of that for non-relativistic motion.

Figure 4
(Click on image to enlarge)

This hemispheric model of the motion of an orbiting object is useful because it allows us to visualise the orbital path length as being foreshortened by relativity while at the same time the radius of the orbit is unaffected by relativity.  The orbital geometry is non-Euclidian and in reality all takes place in just one plane. The introduction of this third dimension is just a device to allow us to visualise what is going on.  The orbiting object sees the distance it travels around one orbit as being reduced by a factor Gamma, but nevertheless sees the orbital radius as being unaffected by relativity since this is at right angles to the direction of travel.  Thus we can represent the radius of the orbit as being the distance from a point on the relativistic orbit to the centre of the hemisphere.

The term Actual Velocity has been adopted to describe the velocity of the orbiting object as seen by a stationary observer.  This is easily calculated as the circumference of the orbital path, the equator of the hemisphere (d), divided by the orbital period (t), both measured by the stationary observer.

The theory postulates that there is a velocity term which is affected by Gamma.  This is termed the Relativistic Velocity. This velocity term can be calculated by taking the foreshortened distance around the line of latitude, which represents the orbital path as seen by the moving observer, divided by the orbital period as measured by a stationary observer.  The foreshortened distance around the orbit is calculated as D=d/γ and the orbital period remains the same as for Actual Velocity (t) and hence this Relativistic Velocity is then easily calculated as:

Equation 3

We can use this term directly in calculating the angular momentum of the orbiting object.   Angular momentum is the product of the mass, the velocity and the radius of an orbiting point object.  However the mass of the object is affected by relativity, appearing to increase the mass by a factor Gamma (γ) and so:

  Equation 4

However since for Gamma to take on a significant value vR must be very close to c, the speed of light and so we can substitute c for vR. Also since the angular momentum of an electron in orbit around an atomic nucleus is given by Planck’s constant we can substitute this for l in Equation 4 to give:

Equation 5

In effect we are simply substituting Relativistic Velocity for Actual Velocity in the standard textbook formula for calculating angular momentum.  This is recognising that the orbital velocity is the distance around the orbit as measured by the moving object divided by the orbital period as measured by a stationary observer.

We can of course use this same argument to substitute Relativistic Velocity for Actual Velocity in the formula for centripetal acceleration and hence derive expressions for centripetal and centrifugal forces.  However in the case of centripetal acceleration it is also useful to derive an expression for the relativistic case from first principles.

The formula for centripetal force was first derived by Christian Huygens in 1659 and describes a constant force acting on a body in circular motion towards the centre of the circle.  When combined with Newton’s second law this leads to the idea that a body in circular motion is subject to a constant acceleration towards the centre called centripetal acceleration.

Figure 5
(Click on image to enlarge)

It is customary when deriving the formula for centripetal acceleration to use velocity vectors directly.  Here we take a slightly different approach and use the distance vectors instead.  This is because the in the proposed theory only the distance component of velocity is affected by relativity and not the time component.  In other respects the derivation is the same as that found in many standard texts.

Consider an object in orbit around a point C at radius R.  At a particular instant t the object is at point A and some short interval of time later Δt it is at point P, having moved through an angle subtended at the centre of the circle of Δθ.

The vector representing the distance moved in time Δt is AB and has length d and is tangential to the circle, hence CAB is a right angle.  At t+Δt the object is at P and has a distance vector PQ, also of length d.   We can translate the vector PQ to A forming AD.  The vector BD then represents the distance moved towards the centre of the circle in time Δt.  Note that for as Δθ tends to 0 the line BD tends to a straight line.

Then

Equation 6

And since APC and ABD are similar triangles (for small Δθ)

  Equation 7

And the acceleration towards the centre of the circle is

  Equation 8

Therefore

  Equation 9

Multiplying both top and bottom by R gives

  Equation 10

But since

  Equation 11

Then

  Equation 12

When we take into consideration the effects of special relativity, the situation becomes a little more complicated.   Although the orbital path is foreshortened, as represented by the line of latitude in Figure 6, and hence the circumference of this circle is reduced by a factor Gamma, the radius of the circle is not affected and remains the same as that for the equatorial orbital path.

Figure 6
(Click on image to enlarge)

Figure 6 attempts to show this by introducing a third dimension and using the hemispherical representation developed above. In reality however the radius and the orbital path are co-planar.  It can be seen from Figure 6 that the angle subtended by a short segment of the circumference is less for the relativistic path than for the non-relativistic path.  From Figure 6 it is evident that:

  Equation 13

And

  Equation 14

Figure 7
(Click image to enlarge)

The distance travelled during time Δt is foreshortened by relativity , instead of travelling a distance AB the object only travels a distance A’B’=D in Figure 7.

  Equation 15

Once again the triangles CA’B’ and A’B’D’ are similar and so the distance travelled towards the centre of the orbit E is

  Equation 16

The acceleration towards the centre of the circle is

  Equation 17

Which is also

  Equation 18

Again we can multiply both denominator and numerator by R to give

  Equation 19

Which gives

  Equation 20
  Equation 21

Equation 21 represents a more general case for calculating centripetal acceleration.  When the orbital velocity is low, under non-relativistic conditions, the value of Gamma is unity and the formula can be simplified to the more familiar one shown in Equation 12. Effectively the formula for centripetal acceleration under relativity substitutes Relativistic Velocity for Actual Velocity in the standard textbook formula.

It is the geometry of the triangle AB’D’ which lies at the heart of the argument.  Here it is argued that the length B’D’ is affected by relativity even though it is measured in a direction at right angles to the direction of travel.  This comes about because the lengths of the two sides AB’ and AD’ are both affected by relativity and the triangle must have geometric integrity and so B’D’ must also be scaled by relativity.  If it was not then the triangle AB’D’ would be a very strange triangle indeed.  It would have to be an isosceles triangle in which the third side could be longer than the sum of the two other sides.  The direction of the vectors AB’ and AD’ could not be preserved and there would have to be some sort of discontinuity when Gamma reached a value of 2 or more since it is at this point that the lengths of the sides would no longer add up.  Even in non-Euclidian geometry such a triangle would not be possible and so B’D’ must be scaled by Gamma.

The measurement of time on the other hand can only take place in the domain of the observer, so the moving observer sees his time in his own domain and the stationary observer sees time in his domain.  The two domains are related by a factor Gamma, but from the point of view of direct measurement this is a theoretical connection.  In other words the stationary observer has no direct access to the moving clock and, vice versa, the moving observer has no direct access to the stationary clock.

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