MODERN ALCHEMIST

Black Hole: The Space-Time Experiment

"There was some interest ten years ago in using a black hole to accelerate a probe to speeds at or possibly even above the speed of light (3.00 x 10⁸ m s^-1 ; 3.00 x 10⁵ km s^-1 ).

The objective was to launch the probe at a given distance from the black hole, with two concerns in mind:

The probe had to be close enough so that the acceleration caused by the black hole would be sufficient to achieve light-speed.
The probe could not be too close, for as the probe neared the black hole, the chance of a collision with another object would increase.

The problem was simplified as follows:

At time=0 s, the probe has a Newtonian speed of 100,000 km s^-1.
At time=0 s, the probe is 10,000,000 km from the black hole.
At time=0 s, the Newtonian acceleration caused by the black hole is 5000 km s^-2.
The goal is a velocity of 300,000 km s^-1.

We can calculate how long it will take to reach light speed since we know the initial velocity and the rate of acceleration, which is a constant.

100,000 + 5000t = 300,000

t = 40 s

It will take 40 seconds to reach light speed.

How far will the ship travel in 40 seconds?

At a speed of 100,000 km s^-1, in 40 s, the ship will travel 4,000,000 km.

But, the ship is accelerating. After ten seconds goes by, the speed is higher, and the ship will travel farther over the time frame (t=10 s to t=20 s) than it did over the time frame (t=0 s to t=10 s).

Let's break this problem into intervals, and attempt to reach an approximate solution.

Let's the speed 100,000 km s^-1 to calculate how far the ship will travel in twenty seconds, and then use the new velocity at 20 seconds to calculate how far the ship will travel for the remaining 20 seconds.

Answer for two calculations: 6,000,000 km.

Let's break the problem into four intervals:

Calculate distance traveled from time=0 s to time=10 s using the velocity at time=0 s,
calculate distance traveled from time=10 s to time=20 s using the velocity at time=10 s,
calculate distance traveled from time=20 s to time=30 s using the velocity at time=20 s, and
calculate distance traveled from time=30 s to time=40 s using the velocity at time=30 s.

Let's do the problem using 40 intervals (with the help of a spreadsheet).

This spreadsheets makes use of iterations from left to right, and from top to bottom:

Column A provides time in seconds.

Cell E1 sets the amount of time between iterations.

Each row in the spreadsheet below row 2 is an iteration.
Since we knew it would take 40 s for velocity to reach light spead, and we selected an iteration time of 2 s so that all the rows would fit in the screen window.

Row B provides distance in km.

Row C provides velocity.

For cell C2, we typed in 100,000.
For cells below C2, we took the number in the cell above, and multiplied it by (5,000 * $E$1).
- 100,000 + (5,000 * 2) = 110,000
- With an acceleration rate of 5,000 km s^-2, the velocity increases 10,000 km s^-1 every 2 seconds.

Row D provides distance traveled. This takes the velocity in the row and multiplies it by $E$1.

In calculating the distance for B3, we add the distance traveled in D2 to the distance in B2.

At this point cells B2, C2, D2, and B3 are filled.

C2 is copied to C3, and D2 is copied to D3.

The row (B3,C3,D3) is copied into rows (B4,C4,D4) through (B22,C22,D22).

Only 2,200,000 km remain; the probe travels 7,800,000 km according to this calculation.

We can see with increasing interations, the "answers" becomes higher and higher, and we begin to fear that perhaps the true answer will mean a collision with the black hole prior to achieving light speed.

Several iterations were performed (from 1 to 40 iterations), and the function of the results vs. the # of iterations is shown below:

It appears that the number of is approaching 8,000,000 km.

The spreadsheet was set up to do 1,000 iterations. Result: 7,996,000 km.

The calculus corresponding to this problem is now available.

Black Holes: Portals into the Unknown

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