Did Einstein Ever Prove

[INRM]

That time was a line segment... started at a given point and ended at a given point?

Did he ever prove time travel was impossible?


INRM

[Wowbagger]

Einstein's theories seem to indicate that it is impossible to travel backwards in time.

But, I don't think they indicate a start point or an end point, at least as long as the "cosmological constant" was in place. It was Hubble's discovery that eventually lead to the idea that time had a begining, at what we call "The Big Bang".

Incidentally, I don't think "point" is the right word. The beginning at end of time are more like gradual fade-outs, or something.

[Ziggurat]

Did he ever prove time travel was impossible?

That isn't something that's provable. Just like you can't prove that no pink unicorns exist. But we can be fairly certain, even in the absence of proof, that this is the case.

[Schneibster]

I interpreted this as a reference not to T, the operation of time reversal, but as a reference to closed timelike curves. Closed timelike curves occur in Kerr's vacuum, an exact solution to the Einstein gravity equations that describes an uncharged black hole; the CTCs occur inside the event horizon. This is why Kip Thorne describes the singularity as being the future once you pass the event horizon.

[Yllanes]

I interpreted this as a reference not to T, the operation of time reversal, but as a reference to closed timelike curves. Closed timelike curves occur in Kerr's vacuum, an exact solution to the Einstein gravity equations that describes an uncharged black hole; the CTCs occur inside the event horizon.
Yes.
This is why Kip Thorne describes the singularity as being the future once you pass the event horizon.
I think you are mixing two things here. The singularity being the future has nothing to do with the existence of CTCs. For example, in a Schwarzschild BH the singularity certainly is the future, but there are no CTCs. In other kinds of BH you can actually avoid crashing into the singlarity once you pass the horizon.

[sol invictus]

That time was a line segment... started at a given point and ended at a given point?

Did he ever prove time travel was impossible?


Quite the contrary. An extremely simply solution to Einstein's equations is ordinary empty flat space, but with the time direction periodically identified (think of rolling a piece of paper up into a tube and gluing it that way - only the direction you're gluing here is time).

That spacetime is physically inadmissible since it contains closed time-like curves (i.e. you can kill your grandfather), but it's a perfectly valid solution to the equations mathematically. As often happens in physics, additional requirements beyond the equations themselves are necessary to forbid such things.

[Tez]

Goedel also showed the existence of CTC's in dust filled rotating universes, with no BH's (its homogenous): http://en.wikipedia.org/wiki/G%C3%B6del_metric

[Yiab]

That spacetime is physically inadmissible since it contains closed time-like curves (i.e. you can kill your grandfather), but it's a perfectly valid solution to the equations mathematically.

I've never understood why thoroughly grounded intelligent scientists (like Stephen Hawking) feel the need to disallow time travel from their models of physics.
Just because time travel would conceptually allow things like the grandfather paradox to happen doesn't mean that physics itself would actually allow them to happen.

Here's my vague, bullshot reasoning:
Essentially, if we view time as a spatial dimension and imagine the universe in its entirety laid out in front of us (err... jumbled up in front of us), we can see closed time-like curves. Since the universe is sitting there as a single, static entity (time is internal to the universe), it clearly contains no inconsistencies. Killing your own grandfather before your father was born is clearly an inconsistency, therefore it simply doesn't occur anywhere.
Essentially, the only type of time travel "paradox" which could exist in this kind of universe would be what is typically called a "predestination paradox" - one which challenges the notion of free will, but does not introduce any inconsistencies from a physical perspective.

Essentially, the only thing this kind of universe would rule out would be contra-causal free will, which I think most of physics rules out anyway, so what's the problem?

[sol invictus]

I've never understood why thoroughly grounded intelligent scientists (like Stephen Hawking) feel the need to disallow time travel from their models of physics.
Just because time travel would conceptually allow things like the grandfather paradox to happen doesn't mean that physics itself would actually allow them to happen.


They disallow them because they don't describe our world.

If you've studied some physics, you know that time translations and energy are related in the same way that momentum and spatial translations are. If you live in a world where one of the spatial directions is compact, then only a discrete set of momenta in that direction are allowed (think harmonics of a stretched string). A similar thing happens when time is compact - only a discrete set of energies are allowed: E_n = n/T, in natural units where T is the periodicity of the time direction and n is an integer.

What this means is that all physics is exactly periodic with period T, or T/2, T/3, etc. Such a theory satisfies the requirement you suggest above - it doesn't have any paradoxes - and this condition on the energies is necessary and sufficient to guarantee that.

The problem is, a theory with evenly spaced energies is what physicists call a "free" theory - like a simple harmonic oscillator. Theories in which there are no interactions at all have this property, and in general such theories aren't very interesting and don't describe the real world.

I should add that if T is very, very, large - larger than the current age of the universe, around 14 billion years - the effects of this might not be so severe, and in that case it might be consisitent. However most of the time these solutions occur the closed timelike curve isn't very "long" - e.g. T is small - and then you really have a problem.

[Yiab]

If you live in a world where one of the spatial directions is compact, then only a discrete set of momenta in that direction are allowed (think harmonics of a stretched string). A similar thing happens when time is compact - only a discrete set of energies are allowed: E_n = n/T, in natural units where T is the periodicity of the time direction and n is an integer.

I thought that, when viewing the universe in general relativity, it was most appropriate to view it as a finite, unbounded 4-manifold, in which case viewing only one dimension at a time would likely not give a good picture of the overall structure of space-time. If time is separated out as independent of the spatial dimensions in this 4-manifold, then I can see why it should be separated from spatial dimensions when talking about it, but if we don't have that independence of dimensions then we can still get the the time-wise slices (technical: regarding the spatial dimensions as a function of the temporal dimension) to be periodic while maintaining its equivalence with spatial dimensions.

They disallow them because they don't describe our world.

If the mathematical models of our world already disallow time travel then why does Stephen Hawking feel the need to create a "chronology protection conjecture" (http://en.wikipedia.org/wiki/Chronology_protection_conjecture)?
If we don't get the non-existence of time travel from the mathematical models of our world, how can we say it is disallowed? After all, it seems clear that time travel could only occur under extreme conditions (I remember reading about a construction of a black hole rotating quickly around a white hole, with the black hole "sewn" to the white hole to form a pathway allowing time travel back to at most a little while after the creation of this structure. I also vaguely recall hearing about a cylinder with the density of a neutron star, 1/4 mile in diameter rotating at 3/4 the speed of light seeming like it would function as a time machine).

As you may be able to tell, I have only a layman's understanding of physics, but I try to pay attention. I do, however, have a reasonable working knowledge of topology.

[sol invictus]

I thought that, when viewing the universe in general relativity, it was most appropriate to view it as a finite, unbounded 4-manifold, in which case viewing only one dimension at a time would likely not give a good picture of the overall structure of space-time.

I'm not sure what you mean by "finite, unbounded 4-manifold". The manifolds are of Lorentzian signature, and they may or may not be finite or bounded. In any non-singular Lorentzian manifold (with signature (1,3) for example), there is always locally an unambiguous separation between time and space. Problems with that might arise globally when there are CTCs, but in the simple example I had proposed there is no such ambiguity.


If we don't get the non-existence of time travel from the mathematical models of our world, how can we say it is disallowed?

Easily - we augment the models with additional requirements. They're our models, after all!

As for chronology protection, it just a conjecture, motivated by the fact that if you start with decent initial data and a region containing CTCs forms you have major problems. For example, quantum corrections probably go out of control at the edge of the region, leading to infinite back-reaction and invalidating the solution. Hawking's conjecture is that good initial data never leads to such a situation, or if it does the problematic region is behind a horizon and therefore doesn't affect anyone outside. I'm aware of one potential counterexample, but it's probably wrong.

[INRM]

Some questions to everybody here...

What's Einstein's "Cosmological Constant"?
-Did Einstein predict an end of time? If so, how did he not predict a beginning?
-What do you mean like a fade-out, all the stars dying? Or like it simply fading out of existance?
-What evidence suggests he's right?

What's a Lorentzian signature?

What's finite, unbounded 4-manifold?


INRM

[sol invictus]

What's Einstein's "Cosmological Constant"?

A term you can add to Einstein's equations. It represents a vacuum energy - a constant energy and pressure density which doesn't depend on the volume or on time. So as the universe expands, the total energy due to the cosmological constant increases (because the density is fixed, but the volume is growing, and energy is energy density times volume). That leads to run-away, forever-accelerating solutions (such as the one we appear to live in).


-Did Einstein predict an end of time? If so, how did he not predict a beginning?

Some solutions have a beginning but no end, some have an end but no beginning, and some have neither.

-What do you mean like a fade-out, all the stars dying? Or like it simply fading out of existance?

If the universe continues to expand, eventually all stars will run out of fuel and the universe will approach a cold state in which the entropy is maximized and (almost) nothing ever happens. Brrr!


-What evidence suggests he's right?


Lots. Try searching "evidence for general relativity" or something.


What's a Lorentzian signature?

That's a little harder to explain... it means a space in which the metric (the function that tells you how far apart two points are) is not positive definite, so that distances (more accurately, the square of the distance) can be either positive, zero, or negative. (Contrast that to a Riemannian signature manifold, in which all distances are positive.) A common convention is to let events separated in time have a negative distance squared, while events separated in space have positive distance squared, and those on the boundary (the light-cone) have zero distance between them.

What's finite, unbounded 4-manifold?

Well, 4-manifold means a space with 4 dimensions. Unbounded might mean without boundary (for example, the surface of a sphere is an example of a 2-manifold without boundary), or often it can mean infinite volume. Finite can mean finite volume, or it can mean with boundary. As you can see, "finite, unbounded" doesn't mean much.

[Wowbagger]

-What do you mean like a fade-out, all the stars dying? Or like it simply fading out of existance? In the context of what I was saying in my first reply to this thread:

We tend to measure time in descrete units, so we are also inclined to think that if time were to ever begin and/or end, there would, logically, be a first and/or last point in time.

But, that does not necessarily need to be the case. If time is continuous in reality - meaning time does not really flow in descrete units, (we merely invent such units for our own reference) - then logically, there could be no single "point" at which time begins and/or ends. In that case, if time were to begin and/or end, anyway, it would have to do so in a gradual, continuous manner, where you could never determine exactly at what "point" it would happen.

This is a little difficult for humans to comprehend, because we think in discontinuous ways. But, nature is not obliged to be easy to understand.

[INRM]

How can you have no beginning but an end?

INRM

[sol invictus]

How can you have no beginning but an end?

INRM

Why not?

Imagine a ray (a line starting at a point ond going off to infinity). You can either think of that as starting at that point and never ending, or as ending at that point but never starting. Same thing, just different interpretation.

That works for time, too.

[The Man]

What Einstein demonstrated is that time and space are different aspects of the same thing, space-time and that all motion in space-time is relative. So the perceptions of forward in this space or backwards in that time are only the results of the frame of reference used. Is it that we are moving forward in time or that time is moving backwards past us?

A simple example, as I walk from my car to the building which I work, in the reference frame of the parking lot I am moving away from my car and towards the door. From my own personal reference frame however, the motion of my feet moves the parking lot reference frame and the door comes closer to me as my car moves further away. From a temporal point of view within my reference frame the door is moving closer to me in time as my car moves further away. If almost at the door I realize I left my ID badge in the car I must then spin the parking lot reference frame on my axis and begin moving it in the opposite direction. As my car comes closer to me both spacially and temporally the door of the building moves further away in both space and time, although it is unfortunately still in my future.

This example demonstrates some aspects inherent in being a self knowing reference frame. Since I only have one front and one back I really can’t change my orientation within my own reference frame but I can rotate other reference frames around me so whatever I need to get to is in the direction that my body is optimized to travel. Similarly by manipulation of other references frames I can ensure that some elements are in my future and others remain in my past.

This simple example however belies one of the unique aspects of time that is unlike the other three spacial dimensions, at least from our perceptions. When considering events with no spacial and only temporal context I can nether change the direction or rate at which those events are temporally moving within my reference frame. For example my next birthday is always moving closer towards me as my last birthday is always moving further away, at the same rate. This directional aspect of time is not reflected in the structure of the laws of physics and is generally considered to be an aspect of our perceptions. Is it that time is moving forward with increasing entropy from some big bang or is that just our perception; perhaps time is actually moving backwards with increasing entropy from some big crunch? Is time moving at all, since the present is the only perception of time we really experience?

At least as far as electromagnetic radiation is concerned there is no preferred direction in time. Maxwell’s equations have both retarded (forward time) and advanced (reverse time) solutions in all directions. However it was the dominance of retarded solutions in our perception of interactions that resulted in the development of Feynman-Wheeler absorber theory. In this theory a particle interaction is comprised of a half amplitude retarded wave from the emitter and a 180 degree out of phase half amplitude advanced wave from the absorber resulting in the full amplitude retarded wave of the particle from emitter to absorber. From this we might surmise that the present is made of equal portions of both the future and the past.



An interesting link on the “Arrow of time”
http://mist.npl.washington.edu/npl/int_rep/VelRev/VelRev.html

[INRM]

So it's possible for there to be NO creation but an end? Wouldn't time have to be going backwards for that to happen?

Which theories seem to be correct though honestly -- No end but a beginning, no beginning but an end, neither, an end and a beginning?


INRM

[Loss Leader]

Did he ever prove time travel was impossible?




The Vulcan Science Directorate has determined that time travel is impossible.

[Yiab]

Well, 4-manifold means a space with 4 dimensions. Unbounded might mean without boundary (for example, the surface of a sphere is an example of a 2-manifold without boundary), or often it can mean infinite volume. Finite can mean finite volume, or it can mean with boundary. As you can see, "finite, unbounded" doesn't mean much.

Everything there is correct up until the last sentence. You actually have given a perfect example of a finite, unbounded 2-manifold: the surface of a sphere - it has no boundary and it has a finite area. A finite, unbounded 4-manifold is a 4-dimensional space without boundary (there's no place you can go where you just can't continue in some direction) but with finite volume.

Admittedly I was somewhat imprecise in my terminology. I should have said "without boundary" instead of "unbounded", and I should have said "of finite width" instead of "finite".

The surface of a sphere meets both of these criteria - $\{(x,y,z)\in \mathbb{R}^3 : x^2+y^2+z^2=1\}$\\
The rotation of $y = e^{x^2}$ about the $z$ axis meets the "without boundary" condition but not the "finite width" one (it has points arbitrarily far apart) even though it has finite volume.\\
A filled-in circle (i.e. $\{(x,y)\in\mathbb{R}^2 : x^2+y^2\leq 1\}$)meets the "finite width" condition but not the "without boundary" one - its boundary is $\{x,y)\in\mathbb{R}^2 : x^2+y^2=1\}$.

[geni]

So it's possible for there to be NO creation but an end? Wouldn't time have to be going backwards for that to happen?

Probably not. That solution can be thought of as a universe that is heading towards a big crunch but where time stretches an infinite degree into the past.


Which theories seem to be correct though honestly -- No end but a beginning, no beginning but an end, neither, an end and a beginning?


INRM

For the universe we are living in it would appear that it has a begining and no end. However depending on certian elements of quantum physics we end up with a cold dead universe full of radation after 10⁶⁴ years (when the last black holes die)

[sol invictus]

Which theories seem to be correct though honestly -- No end but a beginning, no beginning but an end, neither, an end and a beginning?

We know there was a big bang in our past, which in Einstein's theory is a singularity and hence the beginning of time. Current observations indicate that there will not be an end and the universe will expand forever, but that's only an indication: without infinite measurement accuracy (which is impossible) we can't be certain of that.


Admittedly I was somewhat imprecise in my terminology. I should have said "without boundary" instead of "unbounded", and I should have said "of finite width" instead of "finite".


In that case, fine. But again, in GR there's no reason for the 4-manifold to be either without boundary or finite volume. If it has a boundary you need to supplement the Einstein-Hilbert action with a Gibbons-Hawking boundary term.

[INRM]

Probably not. That solution can be thought of as a universe that is heading towards a big crunch but where time stretches an infinite degree into the past.

But wouldn't a big crunch produce a singularity? And don't singularities decay? I'm missing something aren't I?

The answer I would guess would lie in entropy...

INRM

[Buckaroo]

If you live in a world where one of the spatial directions is compact, then only a discrete set of momenta in that direction are allowed (think harmonics of a stretched string). A similar thing happens when time is compact - only a discrete set of energies are allowed: E_n = n/T, in natural units where T is the periodicity of the time direction and n is an integer.


What do you mean by a "compact" spatial or time direction? Is this analogous to constraining a particle in QM, giving rise to quantized energy or momentum? How does that relate to what Yiab was asking?

I'm not challenging your assertion -- I just don't recall such terminology from undergraduate physics, and I'm curious how it applies here. My curriculum didn't hit GR, unfortunately, so what I know of it I've had to scrape together from hither and yon... :)

[sol invictus]

What do you mean by a "compact" spatial or time direction? Is this analogous to constraining a particle in QM, giving rise to quantized energy or momentum? How does that relate to what Yiab was asking?

It means you're only allowed to consider functions that are periodic in that direction. Imagine taking a piece of graph paper and rolling it up. Where before each coordinate pair (x,y) represented a distinct point, now each point has a whole series of coordinate pairs corresponding to it: (x,y), (x+L,y), (x+2L,y), etc., if the x direction is rolled up and L is the circumference. So the only good functions of (x,y) are those invariant under x->x+L.

Remember the infinite square well in QM? That's just a free particle, but you require that the wavefunction vanish at the boundaries of the well, which quantizes the allowed momenta and energies. Now impose a periodic boundary condition instead. Again, the momenta and energies are quantized, in a very similar way. That represents a particle moving in a single compact direction. So yes, they're analogous.

[geni]

But wouldn't a big crunch produce a singularity? And don't singularities decay? I'm missing something aren't I?

The answer I would guess would lie in entropy...

INRM

The problem is that we don't know what happens if both space time and matter collapse completely down to zero. Our theories kinda fall apart.

[Buckaroo]

It means you're only allowed to consider functions that are periodic in that direction. Imagine taking a piece of graph paper and rolling it up. Where before each coordinate pair (x,y) represented a distinct point, now each point has a whole series of coordinate pairs corresponding to it: (x,y), (x+L,y), (x+2L,y), etc., if the x direction is rolled up and L is the circumference. So the only good functions of (x,y) are those invariant under x->x+L.

Remember the infinite square well in QM? That's just a free particle, but you require that the wavefunction vanish at the boundaries of the well, which quantizes the allowed momenta and energies. Now impose a periodic boundary condition instead. Again, the momenta and energies are quantized, in a very similar way. That represents a particle moving in a single compact direction. So yes, they're analogous.

Ah! I think I understand. Thanks! So, in addition to being an argument against CTC's, doesn't this also argue against a "closed" universe? If so, why was such a thing ever considered?

[sol invictus]

So, in addition to being an argument against CTC's, doesn't this also argue against a "closed" universe? If so, why was such a thing ever considered?

Not exactly. First of all, closed universes are really, really big, and so the discretization of momenta is far below anything we can measure. Secondly, their size changes with time, which complicates the analysis. And third, even if the spatial directions are finite volume and don't depend on time that still doesn't mean the energy levels are evenly spaced, because in an interacting theory the energy isn't determined only by the momenta (as it would be for a free particle).

[drzeus99]

That isn't something that's provable. Just like you can't prove that no pink unicorns exist. But we can be fairly certain, even in the absence of proof, that this is the case.

You certainly can prove a negative. I can prove that 1-1 is NOT 3 by proving that 1-1 will ALWAYS be 1. (and let's not play games, since we all know that 1-1 =0. I'm not about to whip out backing proof.)

Because it will ALWAYS be 0, that infers that 1+1 can NEVER be 3.

I'd call that proof, no?


Cheers,
DrZ