Spinning bodies and their effect on space/time.

[Mordred]

I'm not sure what your getting at Sorcerer but if your measuring a volume. You measure it at a point in time. If you tried measuring that volume over a length of time adding up each volume/moment. Everything in the universe no matter how big or small would have infinite volume.

 

 

That's simply not correct. When you ask how much space something occupies it is the volume at the time of the measurement.

[Sorcerer]

I'm not sure what your getting at Sorcerer but if your measuring a volume. You measure it at a point in time. If you tried measuring that volume over a length of time adding up each volume/moment. Everything in the universe no matter how big or small would have infinite volume.

 

 

That's simply not correct. When you ask how much space something occupies it is the volume at the time of the measurement.

Yes, that's exactly what I was getting at. The example given where time switches places uses an eternal black hole and allows for the complete timeline, to infinity of that BH. Since when measuring space (volume, length, anything), we only measure at one point in time. The infinite nature of the timeline of the BH is irrelevant. Thus when we switch the dimensions as we cross the horizon, and we measure a length of time, we should only be doing so over a finite moment in space. The infinite length of 1 or 3 dimensions of time isn't relevant to a measurement of the finite size of 3 or 1 dimensions of space.

 

The article I was referencing is here: http://www.einstein-online.info/spotlights/changing_places

But however if it is demanded that we make either time or space infinite, we must account for the measurement of the corresponding space or time being summed with each corresponding moment/distance so as that too becomes an infinite dimension. We must also be aware that the limits we place on a minimum size for any moment/distance must correspondingly apply to the other half of coordinates. Accordingly, eternal or not, if we choose any two numbers and allow for the minimum to be the infinite series of coordinates between them, infinities can be summed over any length of time or space. Which is a version of Zeno's paradox.

 

Funnily I noticed this stickied on the homepage, I haven't read it yet, but I skimmed over Aleph0 and cardinality briefly about 2 months back. http://blogs.scienceforums.net/pengkuan/2015/12/10/continuous-set-and-continuum-hypothesis/

Edited January 9, 2016 by Sorcerer

[MigL]

I'm really not sure where you get this idea of time and space co-ordinate switching Sorcerer.

Time ( and space ) proceed as normal for you as you approach and cross the event horizon of a Black Hole.

What does happen, is that the light cone, separating time-like and space-like travel 'tips' over on its side ( towards the BH ).

This makes the only possible place you can travel to, the singularity, as that is the only 'place' in your 'future'.

 

As for the Kerr BH, I believe the inner horizon is the one 'modified' by angular momentum, while the outer is accounted for by the mass. And if the inner horizon is enlarged enough, by supplying enough angular momentum to the BH, then both are nullified, leaving a bare 'singularity'.

Keep in mind I'm not exactly sure about this as its been a while and I don't feel like doing the 'research' at this time.

Maybe one of the other members ( Mordred ? ) can clarify.

[Sorcerer]

I'm really not sure where you get this idea of time and space co-ordinate switching Sorcerer.

Time ( and space ) proceed as normal for you as you approach and cross the event horizon of a Black Hole.

What does happen, is that the light cone, separating time-like and space-like travel 'tips' over on its side ( towards the BH ).

This makes the only possible place you can travel to, the singularity, as that is the only 'place' in your 'future'.

 

As for the Kerr BH, I believe the inner horizon is the one 'modified' by angular momentum, while the outer is accounted for by the mass. And if the inner horizon is enlarged enough, by supplying enough angular momentum to the BH, then both are nullified, leaving a bare 'singularity'.

Keep in mind I'm not exactly sure about this as its been a while and I don't feel like doing the 'research' at this time.

Maybe one of the other members ( Mordred ? ) can clarify.

The link is right there..... I can cut and paste the text, but there's some interesting diagrams too. They represent space in 2D and make the 3rd dimension time so it's easier to visualise, it's just that the assumption of an eternal black hole ruins it all. http://www.einstein-online.info/spotlights/changing_places

 

 

 

Through an analogy darkly

So much for space. But how do the new rules apply to our analogous space-time-picture? We use the same guiding principle as before: The direction to which the one-way rule applies will correspond to time, all other directions will be directions of space. Thus, outside the cylinder, the axis direction must again be identified with time. From the outside, the cylinder looks like the boundary of some region of space: At each moment in time (at each constant height along the axis), space, represented by a plane at right angles to the axis, has an inside boundary defined by the cylinder. In the following illustration, one such plane is shown (the one that has always been included in our images before), and the inner boundary is indicated in red:

But inside the cylinder, matters are radically different. The time direction - the one-way direction - now corresponds to the radial direction, while the axis direction is just another unrestricted space direction. In a way, space and time have changed places!

This has remarkable consequences for the way that objects move as time passes. Outside the cylinder, everything is as it was before. But once an object crosses over into the cylinder, it is trapped. Remaining at a constant radial value or leaving the cylinder is as impossible for an object as stopping time or travelling into its own past. Once an object has crossed into the region of space bounded by the cylinder, it can never leave. The cylinder is the boundary (the horizon) of a black hole!

As inexorably as time passes, any object must progress towards the central axis. However, as the axis is reached, matters go wrong. Once the object has reached the axis, it can go no further, but neither can it remain where it is. Any continuation will lead it outward, and outward motion - motion backwards in time - is strictly forbidden. Neither is remaining on the axis an option - inwards motion is compulsory! The axis is the analogue of the black hole's singularity, where our description of space and time breaks down, both in this simple model and in the more complete description within Einstein's general theory of relativity. As they reach the singularity, objects somehow leave the stage (for more information about singularities, see the spotlight textSpacetime singularities).

Inside a black hole: There's no point

Our analogy is useful in understanding one feature of a black hole singularity that is easy to get wrong. If you hear about a spherically symmetric black hole, bounded by its horizon and containing a central singularity, you are likely to picture a cross-section of the black hole which looks like this:

Here, the circle is meant to represent the horizon, and in the center of the black hole, there is a point - the singularity.

In our three-dimensional model, this picture can be obtained by looking at a plane that is orthogonal to the axis: The intersection of the plane with the horizon-cylinder is a circle, the intersection with the singularity-axis is a point. So is this a snapshot of a black hole, showing its interior structure?

Not quite. Only outside the cylinder does the intersection with a plane at constant height ("at constant time" as seen from the outside) correspond to a snapshot. Inside the cylinder, time and space have switched places. Inside, the intersection image doesn't show a snapshot - it shows something much more weird: a caleidoscopic combination of many different times. After all, inside, time is not the axial, but the radial coordinate, and all the different distances from the "center" which you see in the sketch correspond to different moments in time. Instead of the spatial structure of the black hole, the sketch shows a strange mix of space and time!

Likewise, if you think about the unstoppable collapse of a body to form a black hole, you might think that the body ends up with all its matter concentrated in a single point of space - the singularity. But again, this picture of the spacepoint-singularity residing in the center of the black hole is simply wrong. Using our analogy, you can see why. The singularity is the whole of the axis - and the axis represents a space direction. Hence, the singularity is not a point in space - it is infinitely extended!

This is exceedingly weird. From the outside, the region of a black hole looks like the surface of a sphere (in our model with two space dimensions and one time dimension, like the circumference of a circle). But inside that sphere, which has only a finite surface area, you can "hide" objects that are infinitely large - infinitely extended in space. How does this work? Again, it works because time and space trade places. Our simple scenario corresponds to an eternal black hole - a black hole that has always existed and will continue to exist indefinitely in the future. From the outside, the black hole is infinitely extended in time, but has only a finite size in space. Inside, the tables are turned: Time is only of finite extent (it starts at the horizon and ends abruptly at the singularity-axis), but instead one space direction, the axis direction, is now infinitely long.

If you have a hard time getting to grips with this mixture of time and space, rest assured that physicists have a hard time visualizing it, as well. Luckily, physicists have a language in which the properties of simple black holes can be formulated very precisely - the language of mathematics -, and using this formulation as a guide, it is possible to develop a pretty good intuition about spacetime containing a black hole.

The limits of the analogy

Our analogy isn't perfect. In the analogy, the change-over of space and time happens suddenly, at the boundary. In a more precise formulation, the change-over is more gradual. In a way, the time direction is bent more and more inward as you get closer to the black hole. As the bending is strong enough to prevent any object from moving in any direction but inwards, you cross the horizon.

Also, the black hole pictured above is an especially simple specimen. It is spherically symmetric - a so-called Schwarzschild black hole, after Karl Schwarzschild who, in 1915, was the first to write down the equations defining and describing such a black hole; a special solution of Einstein's equations of general relativity. (However, it took physicists more than fourty years to come to understand the weird spacetime geometry that Schwarzschild's equations imply!) As has already been mentioned, this type of black hole is eternal - it has always been there, and will always be there. More realistic black holes with a definite beginning (for example those produced by the collapse of a massive star) or eternal black holes which rotate all have a somewhat more complex inside structure.

Apart from these qualifications, the analogy holds, and it does capture an essential aspect of a real black hole's spacetime geometry - time and space changing places at the horizon, and some fundamental consequences of that exchange.

I guess the last bit says it,

 

 

s has already been mentioned, this type of black hole is eternal - it has always been there, and will always be there. More realistic black holes with a definite beginning (for example those produced by the collapse of a massive star) or eternal black holes which rotate all have a somewhat more complex inside structure.

[Strange]

First I don't understand how the black hole has a finite size in space.

 

Read what it says again: "From the outside, the region of a black hole looks like the surface of a sphere"

 

The black hole has a well defined radius (proportional to its mass) and therefore has a well defined surface area as seen from the outside. That surface area is finite and proportional to the square of the radius (and therefore proportional to the square of the mass).

 

 

2. If the axis direction inside the even horizon is infinite, but the time direction finite, anything travelling to the center would be travelling at an infinite velocity.

 

I am not sure what they mean by "the axis direction".

 

...

 

Just seen you later post, which explains it. Kinda. You do not travel along the axis to the singularity, so the fact it is infinite is irrelevant. But remember that is an analogy and, therefore, in some sense wrong. I suspect the only way to really understand this is via the math. (One interesting point that I have seen is that if you make any attempt to move away from the singularity - e.g. use your rocket engines - then you will just get there sooner.)

 

Any attempt to understand this applying common-sense notions is bound to failure.