Enlightenment [Book 2: SEKTOR V Trilogy]

Appendix for Chapter 30

[FOR READERS WHO ARE INTERESTED IN THE MATHS, THESE ARE THE CALCULATIONS THE ETs SHARED WITH THE SCIENTISTS.]

The Super Massive Black Hole at the center of the Milky Way Galaxy known as Sagittarius A* has a mass equal to 8.555 x 1013kg

This value is 4.30 million times the mass of Sol, your 'sun'

Using the mass of Sagittarius A*, it is then possible to determine its radius, which also gives us the Event Horizon or the Schwarzschild radius (rsh), defined as:

rsh = 2GM/c2

where M is the mass of Sagittarius A*, G is Newton's constant of gravity (6.674×10-11 Nm2/kg2) and c is the speed of light (299,792,458 ms-1).

Using these values, rsh for Sagittarius A* = 1.270 x 101013m

For perspective, this number represents 8.5% of the distance from the Earth to your Sun...

Animation 2:

Because of the severe curvature of space-time around Sagittarius A* due to its incredible mass, the distance a beam of light has to travel is extreme...

And since the speed of light is constant in any frame of reference, time itself must slow down for an observer near Sagittarius A* (relative to an observer far outside the gravitational field of Sagittarius A*)...

Thankfully, the predictions that arise from Einstein's general relativity allow us to very precisely estimate just how much time slows down, or 'dilates,' due to the effects of enhanced gravitational fields...

The two primary factors that control this gravitationally-induced slowing of time (relative to the passage of time far from a black hole) are:

The absolute mass of the black hole in question and the radial distance of the observer from the center of said black hole...

Animation 3...

Once a black hole's radius is known, objects can remain in stable circular orbits if:

The orbit is greater than one and one-half times The Event Horizon.

For the case of Sagittarius A*, this stable orbit radius must be greater than 1.905 x 1013 m

Using a hypothetical radius of 1.906 x 1013 m and following the equation below, which describes time dilation for objects in a circular orbit about a black hole...

t0=tf (1-(1.5(rs/r)))0.5

where to represents the passage of time for an observer near the black hole;

tf is the passage of time for an observer a great distance from the massive object;

rs is the radius of the black hole or its Shwarzchild radius, and;

r is the radius of the observer as it orbits the black hole

We see that a spacecraft orbiting around Sagittarius A* at 1.906 x 1013 m,

a gravitationally-induced time dilation ratio of 43.658 would be achieved...

Or put another way, 1 year orbiting Sagittarius A* would equal more than 43 years on Earth

Animation 4:

Of course, to remain in such an orbit, tremendous speeds must be maintained...

The orbital period t of an object in a circular orbit at radius r from a black hole of mass M, as measured by an outside observer (far from the black hole), is given precisely by Kepler's third law:

G M t2 / (2 pi)2 = r3

Solving for the orbital period t, we find that the time it would take for our spacecraft to circle Sagittarius A* would be 687.693 seconds, or approximately 11 minutes and 28 seconds...

In one full orbit, the vessel would travel 2 pi x r or 119,696,800,000 m.

This translates to a speed of 173,044,573 ms-1 or 58.06% of the speed of light...

Using these values within the context of the formula below, it is possible to quantify the contribution of time dilation due to the speed of the spacecraft in a stable orbit around Sagittarius A*

Tdilspd = 1/((1-(v2/c2)))0.5

The effect of this speed is to add 22.82% dilation, which, when combined with the previously mentioned 43.658 gravitational dilation ratio, yields a total time dilation factor of 53.62.

Or put more plainly, occupants on this ship would experience 1 Earth year while outside observers (not traveling near the speed of light or in close proximity to a black hole) would experience 53.62 Earth years.

This is the dilation factor we are looking to achieve and this is the method we are proposing to achieve it.