Why does time dilation create a velocity barrier but not a density barrier?

[beej67]

Please explain to me what I'm missing:

You can't go faster than the speed of light, because of time dilation. The faster you go, the slower time moves for you as compared to objects at rest, and time would stop for you if you ever went the speed of light. That's what makes C the universal speed limit - time dilation. I get that.

Well, there's also gravitational time dilation. The closer you get to a massive object, the slower time moves for you as compared to objects at rest, and time would stop for you as compared to them if you ever made it to the event horizon. This is an obvious result of the math, and widely accepted since Einstein.

If that's the case, then how can a black hole form at all? Wouldn't time dilation create a "density limit" just as much as it creates a "speed limit"? The math for black holes is valid, but how does one form at all, if the collapsing star time dilates as its density approaches that which would create an EH?

The equations for describing time dilation are effectively the same in both instances, are they not?

Answer phrasing: I've got a masters degree in fluid mechanics, so I'm math competent. I had optics/modern physics in undergrad, but that was 15 years ago. I still know quite a bit of calculus because I teach as an adjunct, but get a little foggy when it gets to heavy use of tensors. If you can keep the math to a "smart undergraduate" level I'll follow fine.

 

[darkhorror]

If there is a spaceship moving away from Earth at .99c. From the Earth's frame of reference the ship is moving at .99c. From the spaceship's frame of reference Earth is moving at .99c. So in the Earth's frame of reference the spaceship's time is moving slower the the earth's. In the spaceship's frame of reference the Earth's time is moving slower than the spaceship's.

That all sound correct to you?

 

[beej67]

Absolutely, and I can tell you're trying to set me up for the answer, but I'm still not seeing it, because when the spaceship comes back to earth, more time has elapsed for the Earth than the spaceship. When you bring the clocks back together and compare them, you discover which one was pushing the barrier, and subject to time dilation.

Is that correct, or am I misinterpreting it?

If objects can pass through the EH of a black hole, why can't they exceed a velocity of C? Is it not time dilation that prevents objects from exceeding C? It seems a bit of a double standard to me, to use the "observer doesn't experience time dilation" excuse in one case but not the other.

 

[Bill_K]

The faster you go, the slower time moves for you as compared to objects at rest...

The closer you get to a massive object, the slower time moves for you as compared to objects at rest

To correct these statements, replace "the slower time moves for you" by "the slower your time appears to a nonmoving observer". According to the time you measure (your "proper time") you'll get to the event horizon right on schedule.

 

[beej67]

To correct these statements, replace "the slower time moves for you" by "the slower your time appears to a nonmoving observer". According to the time you measure (your "proper time") you'll get to the event horizon right on schedule.

I see that, but don't understand why can't we use the same argument to discuss exceeding the speed of light. Is time dilation a barrier, or isn't it?

In the case of an observer falling into a black hole, his time appears to proceed normally to him, but the "universe's time" appears to speed up exterior to him, until it reaches an infinite asymptote as he crosses the EH.

In the case of an continuously accelerating observer, his time appears to proceed normally to him, but the universe's time appears to speed up exterior to him, until it reaches an asymptote as he approaches the speed of light. Would not this continuously accelerating observer "exceed the speed of light, right on schedule?"

In the first case, the answer seems clear. He falls as usual and "leaves the universe" when he passes the EH, since he can no longer be observed. Can we not say the same thing of a continuously accelerating observer who exceeds C? Why is one case treated routinely as a boundary through which one cannot travel, and another treated routinely as a boundary through which one can easily travel, when time dilation describes both?

 

[darkhorror]

Absolutely, and I can tell you're trying to set me up for the answer, but I'm still not seeing it, because when the spaceship comes back to earth, more time has elapsed for the Earth than the spaceship. When you bring the clocks back together and compare them, you discover which one was pushing the barrier, and subject to time dilation.

Is that correct, or am I misinterpreting it?

You well it's kind of correct. But to say that Earth wasn't subject to time dilation is not correct. And what do you mean "pushing the barrier", and what does "subject to time dilation" even mean? That wasn't even the example I gave, as all I was talking about was two frames of reference moving at constant velocity to each other. For what you were talking about you have the Earth as an inertial frame of reference and the ship which will have to accelerate/turn around so it's not an inertial frame of reference.

So when you look at my initial question with the Earth and ship moving at constant velocity to each other, which one is "pushing the barrier, and subject to time dilation"?

If objects can pass through the EH of a black hole, why can't they exceed a velocity of C? Is it not time dilation that prevents objects from exceeding C? It seems a bit of a double standard to me, to use the "observer doesn't experience time dilation" excuse in one case but not the other.

How would moving past the EH of a black hole make it exceed C? I wouldn't say it's time dilation that prevents the object from moving faster than C, it's that C is constant to everyone.

 

[beej67]

So when you look at my initial question with the Earth and ship moving at constant velocity to each other, which one is "pushing the barrier, and subject to time dilation"?

Well you can't tell, because there's no way to compare clocks until the spaceship lands, which means it's velocity can't be constant. Is that at the heart of where your going with this?

How would moving past the EH of a black hole make it exceed C?

It wouldn't. But it would require that it move through a point in space where time dilation relative to an observer's rest frame is infinite, would it not?

I wouldn't say it's time dilation that prevents the object from moving faster than C, it's that C is constant to everyone.

I'm not sure this really answers my question, it just forces me to rephrase it. Which I'm happy to do, because my question may not have been clear.

I hop in CAS, the Constantly Accelerating Spaceship. It has a constant acceleration of 1m/s^2, as seen from its own rest frame. I blast off. 9.5 years later (or something longer due to diffeqs) I'm traveling near C when viewed from the Earth's inertial rest frame, and the Earth is traveling away from me at near C in my inertial rest frame. What prevents me from accelerating that last little bit past C in my CAS? Is it not time dilation, the effect of time slowing to me which reduces my acceleration as seen from the Earth's rest frame, even though my acceleration still appears constant to me? Is this not the basic idea? Then I travel all the way across the universe in the next few minutes of my rest frame, taking aeons of the universe's rest frame, because of time dilation. And by the time I accelerate that last meter per second to equal light speed to an observer at rest, the end of time is here. Asymptote. Right?

 

[Naty1]

You can't go faster than the speed of light, because of time dilation

true, You can't go faster than "c"...but what makes you focus on time?? why not say instead:

"you can't go faster than the speed of light because of length contraction"

I'm not recommending either view ...they are incorrect...and are effects not causes.

(Maybe a rough, really rough, analogy is saying: "I can't run any faster than I do because my legs are too short." Such a statement misses as LOT. )

The faster you go, the slower time moves for you as compared to objects at rest, and time would stop for you if you ever went the speed of light.

Only when viewed from another inertial frame...There is nothing in the frame of someone approaching speed c that changes his local in frame perception of time or his local in frame perception of distance. His clock continues to tick normally, radioactive decay proceeds at it's normal rate, and he measures lengths as he did at rest.

(I guess he IS aware that as he continues to burn fuel at, say, a fixed rate, his acceleration decreases as time passes.)


Via the Lorentz transform you can notice two things:

1) time and space are not entirely separate entities but one frame's time gets split into another frame's space and vice versa.

2) there is a notion of "distance" called the spacetime interval which also mixes space and time together and is agreed upon by all reference frames (i.e. is invariant under the Lorentz transform).

So thinking of time and space as independent entities is a Galilean (classical) view not Einstein's view of relativity.

In fact it appears space and time are related in some subtle way(s) we do not yet fully understand. If we really understood them, we'd understand black hole and big bang singularities.

 

[beej67]

true, You can't go faster than "c"...but what makes you focus on time?? why not say instead:

"you can't go faster than the speed of light because of length contraction"

I'm fine with saying that too. However it gets said, I as the dude approaching C in other people's inertial rest frames get my passage of time in their frame monkeyed with so I never crack C in their frame. How you phrase it is unimportant. What's important, in the case of near C velocities, is that it's said somehow, and however it gets said C becomes a speed limit.

I'm not recommending either view ...they are incorrect...and are effects not causes.

I'm unaware anyone really knows a "cause" for this stuff, other than it just works this way.

Only when viewed from another inertial frame...There is nothing in the frame of someone approaching speed c that changes his local in frame perception of time or his local in frame perception of distance. His clock continues to tick normally, radioactive decay proceeds at it's normal rate, and he measures lengths as he did at rest.

I completely understand that, but where he to view all the other clocks he's flying past he'd see them running faster than his. Right? And that's what's described in the Laplace transformation, right?

(I guess he IS aware that as he continues to burn fuel at, say, a fixed rate, his acceleration decreases as time passes.)

I'm skipping fuel, hypothetical spaceships doesn't need it, (hypodrive!) but I would contend that the rate of fuel burn would not change to our observer on his constantly accelerating space ship, because he never notices a change in acceleration. That's a side issue, and unimportant to my question, which I apparently am still failing to convey properly. An observer "at rest" would certainly notice a reduction in acceleration, and a commensurate reduction in fuel consumption, as the spaceship's clock "slows" to match the laplace transform.

Via the Lorentz transform you can notice two things:

1) time and space are not entirely separate entities but one frame's time gets split into another frame's space and vice versa.

2) there is a notion of "distance" called the spacetime interval which also mixes space and time together and is agreed upon by all reference frames (i.e. is invariant under the Lorentz transform).

I get that, but when you run the math, the Lorenz factor approaches infinity as the velocity of the observer approaches C. Right?

So thinking of time and space as independent entities is a Galilean (classical) view not Einstein's view of relativity.

That's what the Lorentz transformation is for, is it not? To translate between two different essentially Galilean frames, and in so doing describe what each observer "sees?"

 

[Nizzeberra]

Here is a short answer: Black holes don't exist, because you cannot create one. But people still don't realize this, and still believes in their existense.

As a body is compressed towards the density limit (as the radius shrinks towards the radius of the EH) time will come to a halt. This in turn means that all processes, as seen by an external observer, will stop. So yes, there is a density limit (for external observers). The only problem is to convince others about it.

As relativity predicts the existense of black holes, it also predicts that it takes infinite time to create one. :)

An interesting artifact is the collision of two of those... Black bodies? Since time stands still, no particles within these bodies should be able to move relative to each other (still as seen by an external observer). Does this mean that they will never be able to merge into a single spherical object?

 

[Naty1]

How you phrase it is unimportant.

no, it is critical for correctness, communicating and understanding. Also to summarize mathematical results.

I'm not recommending either view ...they are incorrect...and are effects not causes.

I'm unaware anyone really knows a "cause" for this stuff, other than it just works this way

.

Also true; but they are effects not root causes.

but I would contend that the rate of fuel burn would not change to our observer on his constantly accelerating space ship, because he never notices a change in acceleration.

I am not absolutely positive, which is why I posted "I guess..." but I think the local observer would notice a reduction in the time rate of change of clocks in other frames...other clocks would tick asymptically slower...of course he would not know if THEY were changing velocity or he was...

I completely understand that, but where he to view all the other clocks he's flying past he'd see them running faster than his. Right?

you mean everyone sees time running SLOWER in the other frame.

And that's what's described in the Laplace transformation, right?

Lorentz, yes.


(I guess he IS aware that as he continues to burn fuel at, say, a fixed rate, his acceleration decreases as time passes.)

I get that, but when you run the math, the Lorenz factor approaches infinity as the velocity of the observer approaches C. Right?

length is contracted to approach zero, time slows in the OTHER inertial frame. .

So thinking of time and space as independent entities is a Galilean (classical) view not Einstein's view of relativity.

That's what the Lorentz transformation is for, is it not? To translate between two different essentially Galilean frames, and in so doing describe what each observer "sees?"

I think I know what you mean, but here again phrasing IS critical: Galilean frames involve only linear translation displaement of space...no change in distance, no change in time...they are absolutes in Newtonian physics...

see here:
http://en.wikipedia.org/wiki/Galilean_transformation

Problem is, any observer is unaware of the morphing together of space and time...like you and me...all an observer sees in SR is a change of space separately and time separately...Einstein's THEORY is what links them together as observables which appear distinct and separate. In fact, space,distance,time,energy,mass are ALL linked in some mysterious fundamental way that no one understands completely.

 

[harrylin]

Please explain to me what I'm missing:

You can't go faster than the speed of light, because of time dilation. The faster you go, the slower time moves for you as compared to objects at rest, and time would stop for you if you ever went the speed of light. That's what makes C the universal speed limit - time dilation. I get that.

Well, there's also gravitational time dilation. The closer you get to a massive object, the slower time moves for you as compared to objects at rest, and time would stop for you as compared to them if you ever made it to the event horizon. This is an obvious result of the math, and widely accepted since Einstein.

If that's the case, then how can a black hole form at all? [..]

My take on this: The question is erroneous, time dilation doesn't create a velocity barrier.
Turning your account on its head would be better.

If we specify that measurements are done with the reference system from which you depart, then the faster you go, the slower your clocks tick and the harder it gets for you (needing ever more energy) to accelerate to greater speeds. You would need an infinite amount of energy to reach the "velocity barrier" of c.

Similarly, black holes (if they exist) don't depend on time dilation - gravitational time dilation is a function of the neighbourhood of a black hole. And I suppose that clocks don't stop ticking at the event horizon (see also http://en.wikipedia.org/wiki/Event_horizon ).

Does that answer your question? :tongue2:

Cheers,
Harald

 

[Naty1]

If that's the case, then how can a black hole form at all? Wouldn't time dilation create a "density limit" just as much as it creates a "speed limit"? The math for black holes is valid, but how does one form at all, if the collapsing star time dilates as its density approaches that which would create an EH?

Locally, time passes normally as a star collapses...when density reaches critical density the relative horizon forms instantaneously and isolates the singularity from the outside observer.
A collapsing star does not DILATE...it CONTRACTS to a singularity...obliterating space,time,matter...maybe you mean one type of event horizon dilating as the singularity forms...the ABSOLUTE horizon grows smoothly (Hawking popularized this view) .

[The contraction involves electron and neutron degeneracy pressure opposition if you want to read about some of the actual mechanism. Matter is destroyed as protons and electrons are forced together forming neutrons...they are subsequently crushed from existence as we know it when mass is sufficient for a black hole to form. ]

See this discussion for one:
https://www.physicsforums.com/showthread.php?t=488051&highlight=black+holes

there are LOTS of discussions on this issue in these forums.

In the case of an observer falling into a black hole, his time appears to proceed normally to him, but the "universe's time" appears to speed up exterior to him, until it reaches an infinite asymptote as he crosses the EH.

No, nothing special changes at the event horizon for a free falling observer: It is invisible, undetectable, to a freely falling observer. As his speed increases, time slows outside. If he accelerates to stop his fall, only possible just outside the horizon, the Unruh effect and Hawking radiation fry him in an instant.

The most complete non mathematical discussion of black holes I've found is Kip Thorne's BLACK HOLES AND TIME WARPS...a less extensive treatment is in Brian Greene's THE FABRIC OF THE COSMOS.

 

[Nizzeberra]

Similarly, black holes (if they exist) don't depend on time dilation - gravitational time dilation is a function of the neighbourhood of a black hole. And I suppose that clocks don't stop ticking at the event horizon (see also http://en.wikipedia.org/wiki/Event_horizon ).

Here is a quote from the link you provided, that kind of contradicts what you just said (depending on if you are the observer or the suicidal one):

Likewise, any object approaching the horizon from the observer's side appears to slow down and never quite pass through the horizon

Besides... Wether you can travel faster than c depends on who you ask, and what point of view you have. So the answer might be "yes, you can travel faster than c, but you will not reach your destination in time anyway". Here is why I claim you can travel faster than c...

Let's say you have a static destination some lightyears ahead. Along the path, there are other static objects with known positions, so you know the distances between them. As you accelerate, all objects will move towards you at speeds increasing towards c (but never exceeding it). However, space will also contract (Lorentz transformation), and the distance between all objects will shrink. As the reasonable person you are, you know that the distances between the objects really are greater than what you can measure, and thus, your "actual" speed (calculated by using the known static distances between the objects) might as well be greater than c.

And here is a paradox (or is it?). When do you arrive? As we see distant object as they were billion years ago, traveling towards them will make them look like they are speeding up, in order to catch up with the present. So... If you travel at a speed close to c towards an object, will it look like the time stands still, or will it in fact speed up?

 

[Naty1]

Nizz:

As relativity predicts the existense of black holes, it also predicts that it takes infinite time to create one.

no.

Black holes don't exist, because you cannot create one

Good evidence black holes DO exist, but no incontrovertible proof yet.
We can't create time either, nor space...that is not contra evidence for existence.

 

[beej67]

As relativity predicts the existense of black holes, it also predicts that it takes infinite time to create one. :)

Finite time for the stuff falling into the black hole, but infinite time for the third party observer, it seems to me. But that's not how it's expressed in any of the books, and the guys who write physics books are smarter than I am, so I'm trying to figure out where my error is.

My take on this: The question is erroneous, time dilation doesn't create a velocity barrier.

Sure it does. (Doesn't it?) Limit of t' as v approaches c:

where gamma = lorentz factor =

Infinity, because the denominator of the Lorenz factor approaches zero. (right?)

Am I doing it right, or have I got my terms muddled? That would seem to indicate that the clock on the CAS approaches "stopped" to a third party as the CAS approaches C. And while the CAS pilot in his rest frame exceeds a velocity of C relative to a fixed rest frame "right on schedule" to him, the fixed rest frame never quite sees it happen, because they never see his clock tick past 9.51 years. (or whatever the "schedule" is)

If I'm doing something wrong in the math, and we need to shift to calculus, I'm fine with taking the discussion that way. It seems my understanding of it may have some fundamental flaws? But I've read in several places that the reason C is the "universal speed limit" is because of dilation, which is expressed in the limit above.

 

[beej67]

No, nothing special changes at the event horizon for a free falling observer: It is invisible, undetectable, to a freely falling observer.

The event horizon is something that only we the third party observers see. I would contend that the speed of light (as seen by a third party) is invisible and undetectable to a constantly accelerating observer too. But that the constantly accelerating observer's clock slows (in relation to the third party) by the lorentz transform, preventing the constantly accelerating observer from ever exceeding C (as seen by the third party). Am I doing the math wrong?

As his speed increases, time slows outside.

Well there's that, (SR) but even if his speed doesn't increase, he's subject to time dilation (as seen by a third party) by virtue of being in the gravity well, (GR) is he not? And the closer he gets to that event horizon, the more time is dilated as seen by the third party, is it not?

The falling observer never notices the EH, but he should notice the rate of change of events (time) to those third parties increasing relative to his own clock, and were he ever to fall as far as the EH, that should reach an asymptote, should it not? It's the same mathematical term that shows up in each: (1-v^2/c^2)^.5 .

If he accelerates to stop his fall, only possible just outside the horizon, the Unruh effect and Hawking radiation fry him in an instant.

Please let's skip the real world implications of the thought experiment and just stick to the hypothetical as a way of understanding the premises.

The most complete non mathematical discussion of black holes I've found is Kip Thorne's BLACK HOLES AND TIME WARPS...a less extensive treatment is in Brian Greene's THE FABRIC OF THE COSMOS.

Do either of those texts address the issue of how matter can collapse past a point in space where time dilation is infinite as seen from a third party's (the universe's) perspective? I would think this would be an important topic to cover, but I have yet to find it covered anywhere, and I'll often sit in Barnes and Noble fishing through the layman physics books hoping they'll cover it, and they never seem to. Everyone seems to try and focus on the properties of black holes once they form, and not on the process of formation.

 

[beej67]

Good evidence black holes DO exist, but no incontrovertible proof yet.

Most of the evidence I've seen is of "extremely massive spots in space." Then these "extremely massive spots in space" are presumed to be black holes. How do you tell from a billion LY away whether that wad of mass is slightly larger than a basketball or is a singularity? Seems to me that both would affect their neighborhoods in similar ways. But that's not really what I'm asking, and I'd rather not get hung up on it, so let's please shelf the issue of whether or not we can see BHs for now, and talk about how we third parties can see an object move past a point in space where its clock stops relative to ours.

 

[harrylin]

[..] Sure it does. (Doesn't it?) Limit of t' as v approaches c:
[..]

I thought that I made clear that the mistake is in the interpretation of the math... Wasn't I clear or didn't you read my explanations in post #12?

In addition to my earlier explanation: such mathematical relationships don't identify cause and effect. If you see x=2y you cannot tell without a theory behind it if:

- x is the cause of y
or
- y is the cause of x
or
- both have a common cause.

That's a basic thing to understand before doing any research.

Harald

 

[pervect]

Well, we're having the usual problems here - even the seemingly off-topic remarks about black holes not being created are a symptom of the usual problem.

The "usual" problem to which I refer is a belief in "absolute time". Unfortunately, saying the words doesn't seem to communicate anything (judging by past experience, at least), because the person who believes in absolute time hasn't really understood that there can be anything else, and saying the words "absolute time" just sort of rolls over them, without the point being understood.

My advice is to think about how to compare clocks, and to draw space-time diagrams of the actual process of clock comparisons. This will address, operationally, the meaning of "absolute time".

There are a couple of ways to compare clocks - the radar method, and the midpoint method. They can both be shown to give the same results.

The radar method says - emit a radar signal at some event P1. Reflect it off a second event, P2. Receive it again at event P3. Then the time midway between P1 and P3 will be "at the same time" as the event P2, according to the observer emitting and receiving the radar signal at events P1 and P3.

The midpoint method says - pick a frame of reference. The frame of reference consists of wordlines of observers moving with constant velocity. Then a signal emitted at the midpoint well be received "at the same time" by other observers in the same frame.

I've got a few diagrams I could post, but generally I find that people don't actually look at them :-(. So I'm going to formulate this in terms of an exercise, and see if I can get some results or interest that way. (I'll be willing to post mine, if someone makes an attempt.)

Exercise:

For a "stationary observer", pick a reference point on the observers worldlline, and draw on the space-time diagram, using either the radar method, or the midpoint method, the set of points that are at "the same time". as the reference point.

Do the same for a "moving observer".

Draw a third diagram (if needed) where the reference point is shared, i.e. the reference point is on the worldline of both the stationary observer, and the moving observer.

Scale your space-time diagrams so that light always moves at 45 degrees for ease of drawing.

Are the two resulting plots - the lines of simultaneity - the set of points that are "at the same time" - the same or different for the two observers, the moving observer and the stationary observer?

If the plots are different, can the underlying concepts be the same?

 

[beej67]

I thought that I made clear that the mistake is in the interpretation of the math... Wasn't I clear or didn't you read my explanations in post #12?

Lets look at that in more detail.

If we specify that measurements are done with the reference system from which you depart, then the faster you go, the slower your clocks tick and the harder it gets for you (needing ever more energy) to accelerate to greater speeds. You would need an infinite amount of energy to reach the "velocity barrier" of c.

I always see my clock tick at a rate of one second per second. I cannot see otherwise. If we specify that "measurements are done with the reference system from which I depart," then the longer I spend in my 1 m/s/s Constant Acceleration Spaceship, the faster their clocks seem to run to me, even though my clock continues to tick at 1 m/s/s. And the amount of energy I would need to maintain that constant acceleration should be the same, when measured in my rest frame, because my clock is still ticking away in my rest frame, and the energy expended is also measured in my rest frame. The Earth's rest frame accelerates with greater speed away from me every second, though, and their clocks seem faster from my point of view. As others have mentioned, I do not even know I'm approaching C in their rest frame, except when they let me know, or when I watch their clock, or when I measure their acceleration relative to me.

When they watch my clock, it seems to slow down to them, and it nears "stopped" as I approach C.

Lets take these two quotes from two different posts together:

Similarly, black holes (if they exist) don't depend on time dilation - gravitational time dilation is a function of the neighbourhood of a black hole. And I suppose that clocks don't stop ticking at the event horizon (see also http://en.wikipedia.org/wiki/Event_horizon ).

In addition to my earlier explanation: such mathematical relationships don't identify cause and effect. If you see x=2y you cannot tell without a theory behind it if:

- x is the cause of y
or
- y is the cause of x
or
- both have a common cause.

So you would contend that objects cannot cross the time dilation asymptote caused by velocity, but they can cross the time dilation asymptote caused by gravity, and the reason for this has nothing to do with the asymptote, but rather is because of how difficult it is to build a constantly accelerating spaceship and how easy it is to fall down a gravity well?

I'm not in a position to argue that answer, but it seems intuitively sloppy to me. Particularly the idea that there's a spot in space where time doesn't pass relative to outside observers, but objects still retain their velocity to those observers as they move through those spots.

FWIW, your wiki article on event horizons does mention this:

"Likewise, any object approaching the horizon from the observer's side appears to slow down and never quite pass through the horizon, with its image becoming more and more redshifted as time elapses."

..which seems to be what I'm getting at. If that's the case, how does the BH form?

 

[beej67]

I promise to view any diagrams you post in exhausting detail, pervect. :) My purpose here is not to argue, it's to figure out what I'm doing wrong.

 

[harrylin]

Lets look at that in more detail.

Originally Posted by harrylin

"If we specify that measurements are done with the reference system from which you depart, then the faster you go, the slower your clocks tick and the harder it gets for you (needing ever more energy) to accelerate to greater speeds. You would need an infinite amount of energy to reach the "velocity barrier" of c."

I always see my clock tick at a rate of one second per second. I cannot see otherwise.

Why do you think that what you can "see" is relevant? This is not QM!

If we specify that "measurements are done with the reference system from which I depart," then the longer I spend in my 1 m/s/s Constant Acceleration Spaceship, the faster their clocks seem to run to me, even though my clock continues to tick at 1 m/s/s.

Your clock only "continues" to tick at 1 cs/s by constantly switching inertial reference systems. Evidently when you make that claim, you are not comparing your clock with the specified reference system.

And the amount of energy I would need to maintain that constant acceleration should be the same, when measured in my rest frame,

Again, that's another reference system; different perspectives result in different descriptions of the same. No "constant acceleration" of you in the system that I specified. And even no "time dilation" of you at all in the system that you specify. :tongue2:

because my clock is still ticking away in my rest frame, and the energy expended is also measured in my rest frame. The Earth's rest frame accelerates with greater speed away from me every second, though, and their clocks seem faster from my point of view. [..]

Ehm no, if you construe a reference system in which you are momentarily in rest, then their clocks seem slower.

[.. ] Originally Posted by harrylin
"Similarly, black holes (if they exist) don't depend on time dilation - gravitational time dilation is a function of the neighbourhood of a black hole. And I suppose that clocks don't stop ticking at the event horizon (see also http://en.wikipedia.org/wiki/Event_horizon )."
"In addition to my earlier explanation: such mathematical relationships don't identify cause and effect." [..]

So you would contend that objects cannot cross the time dilation asymptote caused by velocity, but they can cross the time dilation asymptote caused by gravity, and the reason for this has nothing to do with the asymptote, but rather is because of how difficult it is to build a constantly accelerating spaceship and how easy it is to fall down a gravity well? [..]

It's a riddle to me why you would think such things. If I correctly understand it, clocks that come near a black hole do not stop ticking.

FWIW, your wiki article on event horizons does mention this:

"Likewise, any object approaching the horizon from the observer's side appears to slow down and never quite pass through the horizon, with its image becoming more and more redshifted as time elapses."

..which seems to be what I'm getting at. If that's the case, how does the BH form?

The description there is about the expected effect of a black hole on observations. You seem to interpret this as an effect of observations on the black hole, just as you apparently interpret its effect on clock rate as its cause. :tongue2:

Harald