Trajectories of two tachions moving infinitely fast?

Question

lie in the plane of greenwich meridian and are tangent to the surface of Earth one at north pole, and the other at south pole.

What is the angle between their trajectories at infinity?
(Ignore rotation of Earth)

Answer 1

Assuming the tachyons are in free fall in the earth's field...

Use units of distance in which the radius of the earth is 1 unit and define the initial positions of the tachyons in spherical coordinates as:

(r,theta,phi) = (1,pi/2,0) and (1,pi/2,pi)

The invariant arc distance for constant time and constant theta = pi/2 is assumed to be:

(ds)^2 = 1/(1-a/r) * (dr)^2 + r^2 (d phi)^2

where a is the Schwarzchild radius:

a = 2GM/c^2 = .886 cm = 1.39 * 10^(-6) earth radii.

The equations for the free fall trajectory for a tachyon travelling infinitely fast are:

r^2 (d phi/ds) = J = 1

this is the analog of angular momentum conservation and is 1 by assumption, and:

ds/dr = 1/sqrt[(1 - J/r^2) * (1 - a/r)]

which follows from the previous equation and the definition of arc length. In going from r=1 to r=oo the change in phi is:

\integral d phi
= \int_(J/r^2)(ds/dr) dr
= \integral_1^oo (1/r^2)* 1/sqrt[(1-1/r^2)(1-a/r)] dr

Since a << 1 this can be well approximated as:

= \integral_1^oo (1/r^2)* 1/sqrt[(1-1/r^2)](1 + a/(2r)) dr

Using the trigonometric substitution r = 1/cos(u) -->

= \integral_0^(pi/2)(1 + a cos(u)/2)) du

= [pi/2 + a/2]

In other words the path is bent in the general way an ordinary orbiting object would be so that it ends up going in the direction a/2 radians farther than it would if it were just travelling in a straight line in the absence of a gravitational field (i.e. pi/2). The other tachyon is bent in the opposite sense so the total angular difference in directions is:

a = 1.39 * 10^(-6) radians

(i.e. 2GM/(R*c^2))

Answer 2

. Since tachions - hypothetical particles the speed of which is greater than the speed of light in vacuum - would have infinite mass, they'd attract each other and, boom, Zero angle.
"... Actually, relativity states that it is not possible to accelerate an object up to a speed greater than that of light since this would need to rely on all of the energy of the Universe; in fact, as the speed of the object increases its mass gets greater and greater. On the other hand, relativity does not prevent the possibility that objects exist with a speed greater than that of light, such as in specific reactions where tardions (vc). In such a case, a particle does not need to be accelerated to a speed greater than that of light since it already exists with a speed greater than "c". The only problem with tachions is that these hypothetical particles should posses an imaginary mass that is too strong of an assumption from a physical point of view. Several unsuccessful experiments have been carried out so far with the aim to find tachions. " .
......... "What's also interesting is that there's a Dirac equation for all the
sectors. For the Galilean tardions, the Dirac equation reduces to the
Schroedinger equation. For the tachyons, it corresponds to the Dirac
equation with a chiral mass term.

The well-known no-go theorem on the non-normalizability of tachyons is
also put into perspective. "

Answer 3

There's no answer. The tachyons could not escape the earth's gravity to get infinitely far away. A tachyon's kinetic energy approaches zero as its speed approaches infinity. Even tachyons must have a critical kinetic energy to escape earth's gravity well. They would orbit the center of the earth (assuming they did not interact with matter), losing speed as they fell below the surface.

Answer 4

You seem to tacitly imply that the geometry of the Universe is Euclidean, whereas relativity theory supports Riemannian. As such, they may meet before "reaching infinity." The difference has to do with the ubiquitous fifth axiom.

Answer 5

Since the universe is infinite but bound they should return to their point of origin, and the angle should be 90 degrees.

Answer 6

There is no answer because tachions don't exist.

You probably meant "Tachyons" but they don't exist either, at least not in a way that can ever be proved.