Sophia and the Spherical Tunnel

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Quotes of the Day

In this analogy,
a black hole is a kind of bottomless pit.
What happens if you fall in? ...
If you could somehow survive the gravitational tides and radiation flux,
and (a likely assumption)
if the black hole were rotating,
it is just possible that you might emerge
in another part of space-time
--somewhere else is space,
some where else in time.
--Carl Sagan, Cosmos, Random House, 1980, p 254
The ring of the wormhole
is actually a sphere
the size of the Schwarzschild event horizon
-- Matt O'Dowd, “| Will Worm Holes Allow Fast Interstellar Travel”, 3:55, 2020-04-22, PBS SpaceTime YouTube Channel

Contemplation for the Day

Let us imagine

You are riding your elephant around the inside of a steel spherical shell. The shell is maybe ten meters in diameter and so, as your elephant’s feet travel along the inside surface of the shell, your head remains almost stationary near the center of the sphere. You are wondering if you are about to get a lesson in why centripetal artificial gravity requires very large radiuses of rotation.

Suddenly the space is bathed in a hot blue light and you fear the reflections off the smooth metal surface will cook you, “Just One, have you been travelling through this tunnel for long?” You recognize the many-bodied unison-voice of Sophia, Goddess of Wisdom.

“It doesn’t seem to be a tunnel. There is no exit or entrance. It is a sphere!” you reply.

“Just One, a spherical shell of limited duration is a four-dimensional tunnel in space-time.” Sophia replies impatiently, “and no physical spherical shell has existed for ever nor will any exist forever.”

“Is that so?” you ask.

“Yes, Just One,” Sophia insists.

Then you hear one Sophia-body speak to the others, “Let’s show her by n-1 dimensional analogy.”

You brace yourself and sure-enough suddenly you are swimming through open space. There is nothing as far as the eye can see, except faint glowing green grid lines measuring out all of the directions, up, left, forward, right, back and down, and the reassuring hide of your elephant under your legs and buttocks. And then the Sophia-bodies appear around you bobbing in and out of existence, shrinking and growing as they try to center themselves on the same three-dimensional brane as you. The Sophia-bodies settle down, each to a single size and position in the space around you and then one says, “Show her a circle.”

The circumference of a circle appears in red directly below you. It is drawn in the horizontal plane so that there is no foreshortening from your perspective and it appears as a circle to you.

Sophia speaks in her many-bodied unison-voice, “Just One. A two dimensional being sees a circle as an enclosed ‘space’, with no entrance or exit.” Two large Sophia-body hands appear; one below you and one above you and clap together squashing you and your elephant into one flat being; surprisingly it doesn’t hurt.

Now you see the world as a single line stretching around you in a circle-like horizon with nothing above it or below it; in fact, you find that you can no longer conceive of above or below. The two-dimensional space around you is dark with a grid of green dots to give you reference points. You feel yourself being moved although you can’t fathom who is pushing you given that you can’t see anything anywhere around that horizon-like circle. Then, quite suddenly, the entire horizon is red. You begin moving through the two-dimensional space and bump into the inside of the red circle and ‘recognize it’, in some sense, as the circle they created below you when you were three-dimensional.

“But,” Sophia’s unison-voice continues, “as a three-dimensional being,” you feel a pinprick and then suddenly you are back to your three dimensional selves: you and your elephant. You are inside the circle and moving upward out of its plane. You just catch a glimpse of a Sophia-hand with a large and nasty-looking syringe slipping out of existence below you, “you are quite capable of seeing how you can pass through a circle without ever passing through its circumference.” As she speaks you and your elephant drift up well above the circle and then a large Sophia hand appears above you and gives you a shove and you and your elephant accelerate downward, falling through the circle, without, as Sophia predicted, ever passing through its circumference.

“Like a lion jumping through a flaming hoop.” You say.

“There were no flames, Just One.” Sophia says sounding puzzled.

“I am not certain she is understanding this!” cries one of the Sophia-bodies floating nearby.

Sophia continues in her unison-voice, “Regardless, Just One, imagine a series of circles all with their planes parallel and their centers aligned along a single axis.”

You don’t really need to imagine. You look up and see a series of circles appear above the red one, all drawn in a horizontal plane and all directly above one another and you so that, from your perspective, they appear as a series of concentric circles. They vary smoothly in colour from red through orange to yellow. As you watch, the number of circles increases until they form a solid tube.

A Sophia-hand appears below you and swats you upward so that you and your elephant shoot upward through all of the circles one after another.

Spherical Tunnel by Genevieve Townsend

“As you can see, Just One,” Sophia’s unison-voice continues, “such a construct forms a sort of tube and …” you emerge from the last circle and it is like emerging from the end of a tube, “ the tube has open ends even though all of the circles it is formed from are complete and without breaks in their circumferences.”

“What was the point of all this again?” you cry, exasperated at all of the swatting and shoving and squashing that has been done to you.

“You need to understand how a series of similar boundaries in n dimensions can be a tunnel in n+1 dimensions.” Sophia replies in her unison-voice.

“If you were a two-dimensional being, “ Sophia begins a new speech.

“Oh no, not again!” you say as the two giant Sophia-hands appear again and squash you and your elephant into one flat being again. Just like last time, it doesn’t hurt and you temporarily lose your ability to imagine up and down.

The world appears to you as a circle, a horizon with no sky or ground in between. And you can’t even imagine sky or ground anymore. You wonder if they would appear as arcs of different colours along your horizon. “passing through such a tunnel, “ Sophia continues and your horizon turns yellow, “It would appear as single circle completely bounding you in all directions. Only if the circles changed in some way from one end of the tube to the other, would you be able to tell them apart.” Slowly the horizon darkens turning more and more orange and then more and more red.

“Just One,” Sophia continues, “When you exit the tube, it will be as if the circle ceases to exist.”

Suddenly the circle disappears and you are out in an open plane again dotted with green dots to show both the left-right and forward-backward directions.

“The same applies in four dimensions, Just One.” Sophia continues and you feel the prick and inflate back into two separate three-dimensional beings again, “A spherical shell that exists for a duration in time, will be open-ended at the past and future ends of its temporal extent. All of the temporal parts of the shell will be connected in such a way that the occupant of the shell can pass from one to the other without having to break through the shell. So the occupant can enter the shell at the moment it comes into existence, …” Suddenly you are back inside your spherical steel shell only the inside is painted red. As your elephant walks you notice that the red paint sticks to her feet as it hasn’t had time to dry “… pass through it for the duration of its existence, … “ The paint on the inside of the shell is slowing changing colour from red to orange to yellow. Your elephant’s foot prints disappear as soon as they are formed. “… and then exit it at the moment of its destruction, … “ and suddenly the steel shell is gone and you and your elephant are back in the empty space with its green gridlines. “ … like passing through a tunnel.”

“Of course!” you exclaim, feigning understanding.

Sophia finishes, “That make sense to you, Just One?”

“Certainly!” you cry as you float back to your normal life, perplexedly convinced that a closed steel spherical shell is actually a four-dimensional tunnel.

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