A tortoise at the centre of time

The centre of it all

The circle is the only geometric shape that’s defined by its centre. There’s no circumference until the centre is defined. No radius, no diameter, no…circle. It’s all about the centre.

Yet a ring’s centre of mass, the mean position of the mass of the ring, is at its centre – where there’s no mass. There’s nothing there.

So where is the centre?

The point of existence

I’m reminded of the stationary centre argument. Say a wheel rotates clockwise. Points on the lower half of the wheel move in the negative X direction. Points on the upper half move in the positive X direction.

The centre of the wheel is where the horizonal motion changes from negative to positive on the y-axis, i.e., it passes through zero. Zero velocity at a point on a rotating system makes no sense because in real terms, this argument suggests that the centre of the wheel doesn’t rotate even though it’s in contact with parts that do. We know this is nonsense, and seemingly the only resolution is that the centre doesn’t exist.

According to Wikipedia, Zeno’s paradoxes are a set of philosophical problems that suggest that motion is nothing but an illusion. If this is true, perhaps we can understand what’s going on (or around) within the circle.

The dichotomy paradox, might help explain why our travel along the circle’s y axis to velocity = zero comes with difficulty.

The thought process is that in order to travel the whole distance to the centre of the circle, we must first reach halfway. To reach half way, we must first reach a quarter distance. And before that, an eighth, and so on. In other words, to make a complete journey we must complete an infinite number of smaller journeys, and since it’s infinite, it’s impossible to achieve.

I think finding the centre of the wheel is the dichotomy paradox viewed 180 degrees from the other direction – we travel half way to the centre, then we need to travel a further quarter, a further eighth, a further… and we never reach the centre.

Or in terms of velocity: as we travel into the circle by a unit distance, the negative velocity becomes less negative. We go another step but the decrease in negativity isn’t by so much. More inward travel, and more decrease – but by decreasing amounts. We never reach zero velocity; we never find the centre.

This sounds more like the “Achilles and the Tortoise” paradox. This is where Achilles chases a tortoise, but by the time he’s got to where the tortoise was, that pesky shelled creature has moved on a bit. Achilles needs to cover that distance too, but by the time he’s got there, you guessed it – the tortoise has moved on. In short, despite Achilles’ higher velocity, he never reaches the slow moving tortoise.

Needle in a black hole

Now lets make this wheel 3 dimensional, and just for fun, call it a black hole.

I’ve written before about a giraffe falling into black hole, so for the sake of equal opportunities for tortoises, let’s throw one of them into these gravitational points of singularity.

I assume that because black holes are so dense they must be solid, but I’ll also assume that the shell of a tortoise is made of even denser material so that it can pass through the surface (and beyond) of a black hole. (Their heavy shell might also explain why tortoises walk so slowly – though seemingly able to outrun Achilles.)

The main thing about black holes is that the gravity is so strong that nothing can escape them, including photons which is why they look black (or would if you could see them). There’s another theory that says if you dig a hole through the middle of a planet and jump in it, you’ll undergo gravitational acceleration until you reach the centre, (then there’s no more mass underneath you) and your momentum will be enough so that you’ll carry on traveling through the tunnel and pop out on the other side of the planet. Assuming no frictional losses, you’re velocity will be zero when you reach the surface on the other side of the planet.

(I’m assuming here that there’s no question of defining the centre of a planet here because someone’s just carved it out of the planet to make way for this tunnel…)

So why doesn’t this latter (Newtonian?) theory hold for black holes? If we chuck a tortoise into a black hole, why doesn’t it come zooming out of the other side?

Is it because it cant find the centre? It is after all, infinitesimally small.

Timely consideration

Let’s whizz back to how time flows in and around a black hole. Time flows slowly in strong gravitational fields. (This is the converse of it flowing quicker in weaker fields, such as some 20,000 km above the Earth where GPS satellites whizz round in medium Earth orbit and need time adjustments applied to their time stamps).

So for our adventurous centre-seeking tortoise, it takes ages to find the centre of the black hole.

Mathematically, if speed equals distance divided by time, and time increases (I think this is right – it’s slowing so losing its value relative to an outside observer – please comment below!) then the tortoise’s velocity must decrease.

In other words, it will take ages for the tortoise to reach the centre of the black hole, but this is no problem because tortoises are used to taking a long time to get anywhere anyway.

But there’s a secondary effect – as he approaches the centre, there is less mass beneath him so the gravitational force weakens. And this means that the time dilation effect reduces too – time speeds back up as he falls towards the centre. This gives Mr Tortoise a sense of hope, and because time is passing by quicker, he does in fact reach the centre.

The centre. Where negative turns through zero to positive. Where there’s no mass below you because there is no below. No gravity, no slowing of time, so time whooshes by.

The tortoise experiences time as usual, but for the observer outside, eons pass and we’re lead to believe that he couldn’t find the centre – or couldn’t escape, because it is after all, a black hole.

Does he escape? Does he whizz off into another universe?

I’m not sure. Personally, I think the tortoise is having the time of his life. He’s the centre of the party, and with mass all around him, he’s pulled in all directions. Popular guy.

I wonder if he’ll ever come out of his shell?

Paul