Time dilation = something is wrong

[giordanobruno]

OK, I just made a few very basic calculations to find out the gravitational acceleration of black holes - BASED on our humble observations so far about black holes. You know mathematics(and physics) is many times counterintuitive(that said: you are surprised when you see the result...) - but this time my intuition told me that the speed of attraction...or just the gravitational acceleration of a black hole will be so fast than...you are right...it appears ot be much faster than the speed of light. In fact, even if our assumptions of radius and mass of a black hole are with >100% margin of error - it still appears that the g of a black hole will be fairly high. That said - it appeared in some of the variables I used that sometimes the speed will be thousands of times faster than the speed of light. So...there are many conclusions to be made here.

But basically, it's not even this interesting observational fact which irritates me - it's obvious that even if our black hole observations are really erronous - then still a speed of 0.9999 of the speed of light will be interesting from the point of view that at least time traveling to the future will be possible. What's interesting is the question: what THE HELL happens when the speed is faster?

Based on how Einstein has formulated the time dilation equations - it appears if the speed = c -> you get "complex infinity" and when the speed is faster...you get imaginary number.
For instance, if you try to explain this logically - it means that a time traveller in a machine moving with velocity = c - will literally go infinite amount of time ahead in the future? And...if the velocity is faster - then I can't even come up with a "crazy hypothesis" as to what might happen.

So...the interesting point of time dilation is that it's a "typical Einstein problem" - so to speak, it's relatively(what a word...) easy to be calculated - but almost impossible to be understood logically. The equations used in the time dilation principle - are so basic that even won't surprise any math wizard - but the logic...it's irritating. I can go as far as using vector calculus or writing c++ programs to express the problem here...but that's not the point - the logic will always be the same: It can't be explained what happens when an object is moving faster than the speed of light.


Any logical suggestions will be appreciated.

thanks

 

[Nabeshin]

Black holes, or anything else, will not accelerate massive objects to greater than the speed of light. It's simply impossible. So all this problem with complex gamma factors is quite irrelevant.

And also, it is almost certain that the gravitational time dilation effect of being near a black hole will outweigh any velocity time dilation effect.

 

[giordanobruno]

Black holes, nor anything else, will accelerate massive objects to greater than the speed of light. It's simply impossible. So all this problem with complex gamma factors is quite irrelevant.

And also, it is almost certain that the gravitational time dilation effect of being near a black hole will outweigh any velocity time dilation effect.

Thanks for the feedback...but here are the calculations:

So...the calculations are basic and I might be doing something wrong despite than, since this is more of a hobby of mine (a financial professional he (quant)) - not a pro physicist. But it's unlikely to be erronous.

So the first formula is the famous one used to calculate g, which is:

And the second important formuls is the other alrebraic equation of special relativity:

http://upload.wikimedia.org/math/e/8/8e8943fcbf5e131710d8d04571f428e5.png

Now in the second equation is obvious what would happen when you plug for v a velocity of 500 000 m/s for instance...but more on that later.

Now about g:

((1.9891*10^30)*10^9)/(2.23795616*10^23)
sollar mass*10^9 (mass of a supermassive black hole)/radius of a supermassive black hole = 10 AU^2
G = 6.673 *10^-11
G*((1.9891*10^30)*10^9)/(2.23795616*10^23) = 593097.601161231

That is...if the black hole "spaghetti" you for just one second...it will be with a speed of 593097 metres per second...obviously faster than c.

I am not sure that gravitational time dilation is of big importance here. I will make a few calculations there and get back...

 

[JesseM]

So the first formula is the famous one used to calculate g, which is:

This equation for gravitational acceleration only applies in Newtonian gravity. General relativity is a more accurate theory of gravity whose predictions are close to Newtonian gravity when the curvature of spacetime is low, but when it's more curved (as with a black hole) their predictions become very different. For a basic introduction to GR, click the "general relativity" link on this page: http://www.aei.mpg.de/einsteinOnline/en/elementary/index.html
Some additional pages here: http://www.aei.mpg.de/einsteinOnline/en/spotlights/gr/index.html

And the second important formuls is the other alrebraic equation of special relativity:

http://upload.wikimedia.org/math/e/8/8e8943fcbf5e131710d8d04571f428e5.png

[/URL]
That equation doesn't show up, but in any case you can't assume that equations from special relativity work in general relativity, although general relativity does reduce "locally" to special relativity via the http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html (which implies that in general relativity, no massive object can ever travel faster than light as seen by a local freefalling observer in the same small region of spacetime).

 

[giordanobruno]

This equation for gravitational acceleration only applies in Newtonian gravity. General relativity is a more accurate theory of gravity whose predictions are close to Newtonian gravity when the curvature of spacetime is low, but when it's more curved (as with a black hole) their predictions become very different. For a basic introduction to GR, click the "general relativity" link on this page: http://www.aei.mpg.de/einsteinOnline/en/elementary/index.html
Some additional pages here: http://www.aei.mpg.de/einsteinOnline/en/spotlights/gr/index.html

That equation doesn't show up, but in any case you can't assume that equations from special relativity work in general relativity, although general relativity does reduce "locally" to special relativity via the http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html (which implies that in general relativity, no massive object can ever travel faster than light as seen by a local freefalling observer in the same small region of spacetime).

Here is the equation (speaking of which I hate wikipedia):

 

[JesseM]

Here is the equation (speaking of which I hate wikipedia):

OK, that's the equation for time dilation as a function of velocity in an inertial coordinate system in special relativity. Like I said it'll still work in a locally inertial system in general relativity, but if you want to consider a coordinate system that covers a large region of spacetime outside a black hole you need to use something like Schwarzschild coordinates, and that equation will no longer apply in such a coordinate system (also, in a non-inertial coordinate system like Schwarzschild coordinates you can no longer assume that light has a constant coordinate speed).

 

[giordanobruno]

OK, that's the equation for time dilation as a function of velocity in an inertial coordinate system in special relativity. Like I said it'll still work in a locally inertial system in general relativity, but if you want to consider a coordinate system that covers a large region of spacetime outside a black hole you need to use something like Schwarzschild coordinates, and that equation will no longer apply in such a coordinate system (also, in a non-inertial coordinate system like Schwarzschild coordinates you can no longer assume that light has a constant coordinate speed).

I admit I've only heard before of the Schwarzschild coordinates...now I am going to take a look and try to periphrase them in the context here.

 

[Nabeshin]

That is...if the black hole "spaghetti" you for just one second...it will be with a speed of 593097 metres per second...obviously faster than c.

Two things of note here.

The first and more obvious one, is the speed of light is 300,000,000 m/s, so this figure of 600,000 isn't even close.

The second, is that in special relativity (or general, but here we are concerned with special) you cannot mindlessly apply Newtonian mechanics. That is, if I start with an initial acceleration of 600,000m/s^2, Newtonian mechanics says that in about 500s I will reach c. This is not correct. Your acceleration, as seen from an outside observer, will actually decrease to zero as your relative velocity increases to c. This isn't even to mention the fact that the force of gravity would not be constant in this scenario! If you want, I could show you the "proof" that an outside observer does not see a radially infalling observer go faster than the speed of light, but it seems to me you want to work it out yourself. :)

 

[DaveC426913]

This isn't even to mention the fact that the force of gravity would not be constant in this scenario!

If we postulate a hypothetical black hole large enough, we can still have a huge acceleration while ignoring the relatively tiny gravitational gradient. This doesn't have to actually be so, it just allows to deal with a simple constant acceleration.

That being said:

the infalling object will accelerate as per the Lorentzian formula, getting closer and closer to c but never reaching it, just like it would under any other constant force (such as propulsion).

 

[giordanobruno]

Two things of note here.

The first and more obvious one, is the speed of light is 300,000,000 m/s, so this figure of 600,000 isn't even close.

The second, is that in special relativity (or general, but here we are concerned with special) you cannot mindlessly apply Newtonian mechanics. That is, if I start with an initial acceleration of 600,000m/s^2, Newtonian mechanics says that in about 500s I will reach c. This is not correct. Your acceleration, as seen from an outside observer, will actually decrease to zero as your relative velocity increases to c. This isn't even to mention the fact that the force of gravity would not be constant in this scenario! If you want, I could show you the "proof" that an outside observer does not see a radially infalling observer go faster than the speed of light, but it seems to me you want to work it out yourself. :)

Well...that was some error about the speed :)...but as I mentioned, when I made the calculations earlier - the speed was thousands times over 300 millions m per second (300 000 000)...well ofcoure it's: 299 792 458 for more exact calculation.
Yes, an observer outside won't see it faster...but the time passing here is important.
And if you mean that the object being attracted will change it's own gravity - won't this even increase speed even futher? OK, I can't confirm this one with certainty...but it follows that the time dilation difference is still very much, though this tells nothing about speed of attraction:

How to formulate this one to tell speed? Since we have time and distance...velocity should be easily calculated?

 

[JesseM]

How to formulate this one to tell speed? Since we have time and distance...velocity should be easily calculated?

This equation is meant to tell you the time dilation for a clock hovering at constant radius from the black hole, as compared to a clock an arbitrarily large distance away from the black hole. I think you'd need some more complicated calculation for the time dilation on a clock that was not hovering at constant radius but was instead falling towards the event horizon.

 

[giordanobruno]

This equation is meant to tell you the time dilation for a clock hovering at constant radius from the black hole, as compared to a clock an arbitrarily large distance away from the black hole. I think you'd need some more complicated calculation for the time dilation on a clock that was not hovering at constant radius but was instead falling towards the event horizon.

Some exponential integral or ODE can do the job then? I guess one can even apply a differential equation which assumes an upper limit (like the one used in...zoology predator habitat :) by Lodka) and c being that limit. The problem is...why assume a speed limit of c at first place...

 

[JesseM]

By the way, if you're interested in knowing what the worldline of an object falling into a black hole would look like in Schwarzschild coordinates, see the left-hand diagram on this page from the book Gravitation by Misner, Thorne and Wheeler:

The black line is the falling object's worldline, the cones emanating from various points on its worldline represent the future light cones from those points, so you can see that a light ray sent inwards from any event on the object's worldline always falls in faster (the slope is farther from vertical, the vertical axis being the time coordinate and the horizontal axis being the space coordinate, although the meanings become reversed inside the horizon) than the object itself.

 

[JesseM]

Some exponential integral or ODE can do the job then? I guess one can even apply a differential equation which assumes an upper limit (like the one used in...zoology predator habitat :) by Lodka) and c being that limit. The problem is...why assume a speed limit of c at first place...

The idea is just to assume the basic laws of general relativity, and then apply them to the specific problem of figuring out the proper time for a clock falling into a black hole. If you're asking why we assume the laws of general relativity are correct in the first place, there's plenty of evidence that they give very accurate predictions in a wide variety of situations, if you want some examples just ask.

 

[giordanobruno]

The idea is just to assume the basic laws of general relativity, and then apply them to the specific problem of figuring out the proper time for a clock falling into a black hole. If you're asking why we assume the laws of general relativity are correct in the first place, there's plenty of evidence that they give very accurate predictions in a wide variety of situations, if you want some examples just ask.

Well I know they have been confirmed in many empirical cases...the problem is that it's somehow paradoxical how some people reject the idea that c can't be exceeded simply because we have never seen a higher speed. Yes, but when Einstein created his theories - he obviously didn't try to run at speed closer to c...just to be sure that his theories work. To the best of my knowledge even Lorentz was one of the first people to question the time dilation as an effect. This is why I usually prefer mathematics to physics - physics sometimes relies too much on observation (well...it's better to simply assume that a coin will fall on tail 50% of the time since it has 2 sides - rather than tossing that coin 10,000,000 times...which is still insufficient.).
On a side note, I have some work now...and after that will try to look up some integral that will fit somehow in the idea of an object falling towards the event horizon.

 

[espen180]

This is why I usually prefer mathematics to physics - physics sometimes relies too much on observation

Well, physics is a science, mathematics isn't. How do you expect to predict results of experiments if you don't do observations and measurements?

 

[JesseM]

Well I know they have been confirmed in many empirical cases...the problem is that it's somehow paradoxical how some people reject the idea that c can't be exceeded simply because we have never seen a higher speed.

But that's not why physicists think c can't be exceeded, they think that because it's a clear prediction of relativity that the energy needed to accelerate a massive object to a given velocity v (relative to an inertial frame) would approach infinity as v approaches c. You couldn't change this without totally changing the mathematics of the the theory, and then you'd need to account for why the theory is so accurate in all its experimentally-verified predictions. Also, relativity's predictions about how energy increases as a function of speed can be tested up to a very high fraction of the speed of light in particle accelerator experiments, and since the speed/energy relation is accurate in these cases that's reason to be more confident it will continue to be accurate for arbitrarily high sublight speeds.

To the best of my knowledge even Lorentz was one of the first people to question the time dilation as an effect.

But there is now abundant evidence for the accuracy of relativity's predictions about time dilation (both velocity-based and gravitational) are correct--again, if you what info on the experimental evidence just ask.

and after that will try to look up some integral that will fit somehow in the idea of an object falling towards the event horizon.

Unless you are deriving them from general relativity this seems like a pointless exercise, you can't just guess what the equation would be.

 

[Nabeshin]

This equation is meant to tell you the time dilation for a clock hovering at constant radius from the black hole, as compared to a clock an arbitrarily large distance away from the black hole. I think you'd need some more complicated calculation for the time dilation on a clock that was not hovering at constant radius but was instead falling towards the event horizon.

Well, for a radially infalling particle dropped from infinity the four velocity is (c=G=1):
$$u^{\alpha}=\left(\left(1-\frac{2M}{r}\right)^{-1},-\sqrt{\frac{2M}{r}},0,0\right)$$
So the time dilation factor with respect to the coordinate observer (at infinity) is just the time component. That is to say, clocks at infinity run u^t faster than those at a distance r.

 

[giordanobruno]

Unless you are deriving them from general relativity this seems like a pointless exercise, you can't just guess what the equation would be.[/QUOTE]

not necesserally ...I mean consider this problem which can relate to special relativity -
I am almost sure no one has used this differential equation in the context of relativity - but the idea here is to see how a quantity will change with respect to some other value, at the same time knowing that this quantity has a limit. In other words, one might ask "how will speed change with respect to time, knowing that the speed cannot exceed some value - like the speed of light": if the speed of light is the limit(c-1 meter...note that the equation assumes that the speed of a body can be equal to c...so the trick is to assume that c is not the limit but some amount slighlty below c - such as one meter slower than c) and acceleration is some arbitrary value. Also suppose that we take as possible that speed < c is quite possible so at some time t_0 the speed is taken to be 100,000,000 meters/s. What will be the speed of the body after say 800 seconds? Well this differential equation can be used and solved easily by integrating both parts:

So:

299792457.99/(1 + e^((-0.02*800) +ln(2)))


299792457.99 is taken to be the maximum speed to which a body can accelerate.
e is the 2.17...constant.
0.02 is arbitrary value of 2% exponential acceleration since the closer it moves to the object - the larger the speed due to acceleration.
800 is 800 seconds of travelling.
ln(2) is a constant (well not exact - but shows the idea here...).

In the following example it will take less than 1000 seconds obviously for an object to reach the limit at which it can no longer accelerates...


Only problem is that...the speed of light is not the limit usually :).

 

[JesseM]

not necesserally ...I mean consider this problem which can relate to special relativity -
I am almost sure no one has used this differential equation in the context of relativity - but the idea here is to see how a quantity will change with respect to some other value, at the same time knowing that this quantity has a limit.

The point is that it's fairly meaningless to just pull equations out of the air without defining the physical details of the problem you want to consider, and then figuring out what the theory would actually say about that scenario. If you want to consider acceleration in special relativity, for example, then you should probably consider how the proper acceleration (instantaneous acceleration in the object's own rest frame at a given moment) is varying with proper time (time as measured by the object's own clock), since these are frame-invariant quantities with a clear physical meaning. If a ship is firing a rocket at a constant rate then it will have approximately constant proper acceleration (ignoring the issue that over the long term, firing the rocket burns up fuel and thus causes its mass to decrease, which allows you to get more acceleration from the same force). In the case of constant proper acceleration, you can see the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html for the correct equations to tell you how the rocket's position and velocity will change with time in the frame of an inertial bystander. For velocity, the answer is that v(t) = at / sqrt[1 + (at/c)^2]. Notice that this can be rewritten as c times the dimensionless number (at/c) / sqrt[1 + (at/c)^2], and that this dimensionless number is always less than 1, though it approaches 1 in the limit as t approaches infinity.

 

[Frame Dragger]

Well I know they have been confirmed in many empirical cases...the problem is that it's somehow paradoxical how some people reject the idea that c can't be exceeded simply because we have never seen a higher speed. Yes, but when Einstein created his theories - he obviously didn't try to run at speed closer to c...just to be sure that his theories work. To the best of my knowledge even Lorentz was one of the first people to question the time dilation as an effect. This is why I usually prefer mathematics to physics - physics sometimes relies too much on observation (well...it's better to simply assume that a coin will fall on tail 50% of the time since it has 2 sides - rather than tossing that coin 10,000,000 times...which is still insufficient.).
On a side note, I have some work now...and after that will try to look up some integral that will fit somehow in the idea of an object falling towards the event horizon.

Good luck with that level of rigorous empirical evidence on stellar+ mass Black Holes. Is there any chance that you're mistaking rotational frame dragging effects or apparently superluminal jets with a real FTL phenomenon? I should add, if your love of mathematics is so great, and you accept the empirical evidence which has validated GR, don't you appreciate that those experiments also match the derived value of c?