Confusion about closed timelike curves

[laymanB]

[Moderator's note: split off from another thread since this is a separate topic.]

I am confused about closed timelike curves. To say that they are possible using an exact solution to the EFEs, like the Godel metric, and that they actually correspond to something in reality is where my confusion lies. What exactly does an exact solution to the EFEs tell us? Aren't they constrained by the distribution of matter density and observational facts? Does GR apply to what is possible outside of our spacetime?

Secondly, personally, the idea of something creating itself seems absurd. It seems like logic itself would designate the term "self-creation" as analytically false. Am I missing something?

 

[kimbyd]

I am confused about closed timelike curves. To say that they are possible using an exact solution to the EFEs, like the Godel metric, and that they actually correspond to something in reality is where my confusion lies. What exactly does an exact solution to the EFEs tell us? Aren't they constrained by the distribution of matter density and observational facts? Does GR apply to what is possible outside of our spacetime?

Closed timelike curves certainly are weird, but aren't necessarily impossible. The fundamental problem here is that the notions of cause and effect that we usually think about have no basis in physics. In physics, it's a question of whether the configuration of the system fits the equations. And it is often possible to write down a solution that is a closed timelike curve that fits all of the equations, even though it doesn't make any intuitive sense.

It may be that closed timelike curves are impossible, but there's yet to be any proof presented that shows this.

Secondly, personally, the idea of something creating itself seems absurd. It seems like logic itself would designate the term "self-creation" as analytically false. Am I missing something?

"That doesn't seem reasonable" isn't a good barometer for truth. There are many things about our universe that seem quite absurd to somebody who hasn't learned about the science in detail (and many other things that seem absurd even then).

 

[laymanB]

Thanks @kimbyd I appreciate the responses.

The fundamental problem here is that the notions of cause and effect that we usually think about have no basis in physics.

Could you expound on this a little more? When we model our observable universe with GR, do we see any effects that do not have causes? Or do we see an effect that is its own cause?

And it is often possible to write down a solution that is a closed timelike curve that fits all of the equations, even though it doesn't make any intuitive sense.

That's part of what I'm asking. We can write down exact solutions to the EFEs like Godel's. So, why do we choose certain ones to use in our best models?

It may be that closed timelike curves are impossible

How would you determine them to be impossible?

"That doesn't seem reasonable" isn't a good barometer for truth. There are many things about our universe that seem quite absurd to somebody who hasn't learned about the science in detail (and many other things that seem absurd even then).

Absurd may have been too strong a word. Do any of the known laws of physics describe something creating itself?

 

[kimbyd]

Could you expound on this a little more? When we model our observable universe with GR, do we see any effects that do not have causes? Or do we see an effect that is its own cause?

It's more that the terms "cause" and "effect" have no relationship to anything that exists in the theory at all. General Relativity describes which 4-dimensional configurations of the universe make sense. It says, for example, that no object can outrun a light ray. So General Relativity will say that the configuration at time A is consistent with the configuration at time B. But it says nothing at all about A causing B, or B causing A. In fact, you can make time go in either direction within the theory and it will say the exact same thing about whether or not it's a valid solution to the equations.

The notion of cause an effect can be best understood as a way of understanding the global increase in entropy over time. Lower-entropy states that occurred before are considered to "cause" higher-entropy states that happen later. Sometime in the early universe there was an extremely low-entropy state, and the increase in entropy from that state creates the "arrow of time" that we experience.

That's part of what I'm asking. We can write down exact solutions to the EFEs like Godel's. So, why do we choose certain ones to use in our best models?

That's a complicated question. Ideally, the solutions can be distinguished experimentally from solutions other people have proposed, so that somebody can run an experiment to say which one is correct. If that's not possible, it often comes down to personal choice. Different theorists settle on different notions of what sorts of theories are more or less reasonable, and investigate those theories that they think have more merit. The different decisions theorists make on what is or is not reasonable lead to theorists exploring a diverse array of possible models.

How would you determine them to be impossible?

By showing that there is some contradiction in the math.

Absurd may have been too strong a word. Do any of the known laws of physics describe something creating itself?

In a sense, yes. A simple example is nuclear decay. Every second, a nuclear decay has a constant probability of occurring (depending upon the atomic nucleus involved). The decay itself is made possible by the configuration of the nucleus, but no event causes it to actually start. It just has a random chance of happening every moment. When it does happen, it can be thought of as an uncaused event. There was a physical configuration that made it possible, but nothing actually caused it to happen at a specific moment.

 

[laymanB]

From Carroll's article: http://www.preposterousuniverse.com/blog/2007/04/27/how-did-the-universe-start/

"Personally, I think that the looming flaw in all of these ideas is that they take the homogeneity and isotropy of our universe too seriously. Our observable patch of space is pretty uniform on large scales, it’s true. But to simply extrapolate that smoothness infinitely far beyond what we can observe is completely unwarranted by the data. It might be true, but it might equally well be hopelessly parochial. We should certainly entertain the possibility that our observable patch is dramatically unrepresentative of the entire universe, and see where that leads us."

While I agree with his point about the assumptions of homogeneity and isotropy of the universe, it doesn't seem like he gives credence to the theorems of Borde, Guth, and Vilenkin. (BGV) https://arxiv.org/pdf/gr-qc/0110012.pdf where they say:

"In this section we show that the inequalities of Eqs. (4) and (6) can be established in arbitrary cosmological models, making no assumptions about homogeneity, isotropy, or energy conditions."

"Our argument shows that null and timelike geodesics are, in general, past-incomplete in inflationary models, whether or not energy conditions hold, provided only that the averaged expansion condition Hav > 0 holds along these past-directed geodesics."

I don't know enough to comment on the validity of the theorems, but they seem to show that you are left with a boundary somewhere.

 

[laymanB]

It's more that the terms "cause" and "effect" have no relationship to anything that exists in the theory at all. General Relativity describes which 4-dimensional configurations of the universe make sense. It says, for example, that no object can outrun a light ray. So General Relativity will say that the configuration at time A is consistent with the configuration at time B. But it says nothing at all about A causing B, or B causing A. In fact, you can make time go in either direction within the theory and it will say the exact same thing about whether or not it's a valid solution to the equations.

I am in the very elementary stages of trying to learn GR, but I think that I understand that 4-D spacetime in GR is a static model, the so-called block universe, correct? Is there a way to use the theory to describe the worldline of a particle moving through space as a function of time? Can you model dynamical processes with it?

The notion of cause an effect can be best understood as a way of understanding the global increase in entropy over time.

Is the notion of global entropy well defined? If the universe is spatially infinite, how could we discuss it as a closed system? Do we just define it for a comoving volume of the observable universe?

In a sense, yes. A simple example is nuclear decay. Every second, a nuclear decay has a constant probability of occurring (depending upon the atomic nucleus involved). The decay itself is made possible by the configuration of the nucleus, but no event causes it to actually start. It just has a random chance of happening every moment. When it does happen, it can be thought of as an uncaused event. There was a physical configuration that made it possible, but nothing actually caused it to happen at a specific moment.

I agree that radioactive decay appears to be a a stochastic process consistent with the probabilistic nature of quantum theory, but saying that we have a certain probability when an alpha or beta particle will be emitted is different than saying the particle created itself. We may be able to create many subatomic particles by smashing a couple protons together, but no one (that I'm aware of) is saying those subatomic particles came from nowhere or created themselves.

 

[kimbyd]

I am in the very elementary stages of trying to learn GR, but I think that I understand that 4-D spacetime in GR is a static model, the so-called block universe, correct? Is there a way to use the theory to describe the worldline of a particle moving through space as a function of time? Can you model dynamical processes with it?

Yes, particles moving through space-time is a fundamental component of General Relativity, which states that such particles move along what are known as geodesics.

Modeling dynamical processes is harder, as you need complex interactions between particles, and it's just very difficult to do that in any sort of detail in General Relativity due to the complexity of the math involved. There's no theoretical challenge to modeling such processes. It's just hard to do. But ultimately this question of yours doesn't help clarify the arrow of time situation. No matter what system you come up with, if you reverse the arrow of time, it will still fit the equations.

Is the notion of global entropy well defined? If the universe is spatially infinite, how could we discuss it as a closed system? Do we just define it for a comoving volume of the observable universe?

For an expanding, approximately homogeneous universe, you can just consider the entropy of a comoving volume.

I agree that radioactive decay appears to be a a stochastic process consistent with the probabilistic nature of quantum theory, but saying that we have a certain probability when an alpha or beta particle will be emitted is different than saying the particle created itself. We may be able to create many subatomic particles by smashing a couple protons together, but no one (that I'm aware of) is saying those subatomic particles came from nowhere or created themselves.

It's not logically different. Not in terms of cause-and-effect relationships. It's an effect without a causal event.

In terms of creating particles, conservation laws usually forbid such processes. The exceptions that may allow such things to occur are largely theoretical at this point. But that's a different obstacle than the cause-and-effect obstacle you're talking about. Nuclear decays show that such an objection doesn't make sense in quantum mechanics.

 

[PeterDonis]

4-D spacetime in GR is a static model, the so-called block universe, correct?

It's a description of a 4-dimensional geometry.

Is there a way to use the theory to describe the worldline of a particle moving through space as a function of time?

The 4-dimensional geometry already includes this: the worldline of any particle is a curve in the 4-dimensional geometry. If you make some arbitrary choice of coordinates, that implies a split into "space" and "time", and, provided the coordinates cover the region of the geometry that contains the particle's worldline, you can then use the particle's worldline to write a description of the particle moving through "space" as a function of "time". But this description will depend on your choice of coordinates; there is no unique way to do it.

Can you model dynamical processes with it?

A "dynamical" process in GR is just a 4-dimensional geometry (or a portion of one) like any other; the word "dynamical" really just means "the geometry is more complicated than the simple kinds that are used for pedagogy in textbooks".

 

[laymanB]

In terms of creating particles, conservation laws usually forbid such processes.

It is my understanding that matter and radiation can be created from energy, like the primordial nucleosynthesis in the early universe. But are there any laws of physics that allow the creation of energy or matter without either first existing? This sounds to me too similar to the virtual particles borrowing from the vacuum energy fallacy.

It's not logically different. Not in terms of cause-and-effect relationships. It's an effect without a causal event.

What is the proof that radioactive decay has no causal event? I don't know that much about nuclear physics but it seems that the interaction between the strong nuclear force, the weak nuclear force, and the electromagnetic force within the given configuration of the atomic nucleus would be the cause for such an event as the emission of a alpha or beta particle. Whether we can have the knowledge of such interactions a priori is a separate question.

Nuclear decays show that such an objection doesn't make sense in quantum mechanics.

To me this argument is analogous to saying that we observe a single photon hitting a photographic plate and then we conclude that the photon had no cause. The probability comes in the distribution of the possible locations where the photon will actually leave its mark on the plate. The cause of the photon was the single photon source that sent it through the apparatus. Our knowledge of its properties, position, and momentum are independent of the fact that we generated a single photon from our source.

 

[laymanB]

The 4-dimensional geometry already includes this: the worldline of any particle is a curve in the 4-dimensional geometry. If you make some arbitrary choice of coordinates, that implies a split into "space" and "time", and, provided the coordinates cover the region of the geometry that contains the particle's worldline, you can then use the particle's worldline to write a description of the particle moving through "space" as a function of "time". But this description will depend on your choice of coordinates; there is no unique way to do it.

And if we set up our analysis with this arbitrary set of coordinates that includes the region of geometry that includes the worldline of the particle we are interested in, then does not the ordinary meaning of cause and effect apply within these coordinates? If other worldlines of particles intersect with our target one, do they not alter the trajectory of our target worldline based on classical kinematics? Could you describe a person throwing a ball into the air and catching it on the surface of the Earth with GR using geometry and worldlines?

 

[laymanB]

But ultimately this question of yours doesn't help clarify the arrow of time situation. No matter what system you come up with, if you reverse the arrow of time, it will still fit the equations.

Are you saying that exact solutions to the EFEs that model the universe as a whole make it a necessary consequence that the universe has to have time symmetry?

 

[relatively-uncertain]

It is my understanding that matter and radiation can be created from energy, like the primordial nucleosynthesis in the early universe. But are there any laws of physics that allow the creation of energy or matter without either first existing? This sounds to me too similar to the virtual particles borrowing from the vacuum energy fallacy.

[Moderator's note: split off from another thread since this is a separate topic.]

Secondly, personally, the idea of something creating itself seems absurd. It seems like logic itself would designate the term "self-creation" as analytically false. Am I missing something?

I'm not very advanced in physics(high-school student) so forgive me for any misrepresentations but no I don't know any laws of physics that show energy or matter without either existing first, but I don't suppose you can say that the "universe created itself".. it does seem logically impossible for something to create itself without something existing there first because something can't come from nothing... but 'nothing' in the physics sense, wouldn't that be just a quantum vacuum? So if you had a quantum vacuum to begin with , why can't a universe create itself? (asking did the quantum vacuum create itself wouldn't make sense right...if the argument is that a quantum vacuum is intrinsic to the nature of the universe)

 

[rootone]

you can say that the "universe created itself".

You can say that. but it doesn't mean much.
You can say that 'nothing' is the sum of everything.
0 = 42 -42.

 

[PeterDonis]

if we set up our analysis with this arbitrary set of coordinates that includes the region of geometry that includes the worldline of the particle we are interested in, then does not the ordinary meaning of cause and effect apply within these coordinates?

Not if the region contains closed timelike curves. For example, if it's a region of the Godel spacetime, the worldline of the particle you're interested in could be a closed timelike curve. But the failure of the ordinary concept of cause and effect in that case will be global, not local. See below.

If other worldlines of particles intersect with our target one, do they not alter the trajectory of our target worldline based on classical kinematics?

The worldlines in question will obey relativistic kinematics, yes. (I make this clarification because in this context many people use "classical" to mean "Newtonian", and Newtonian kinematics are not precisely correct.) But it's not correct to think of the collision as "altering" the worldlines, because the worldlines are what they are; the 4-D solution does not "change" when a collision happens. It just includes worldlines that intersect.

Could you describe a person throwing a ball into the air and catching it on the surface of the Earth with GR using geometry and worldlines?

Of course.

 

[PeterDonis]

'nothing' in the physics sense, wouldn't that be just a quantum vacuum?

It depends on which physicist is using the term. The problem with ordinary language words like "nothing" is that there is no one way to translate them into the math used in physics. What's more, in many cases (and the word "nothing" is such a case), no translation preserves the connotations of the ordinary language term that make people want to use it in discussions like this. A quantum vacuum is not "nothing" in the sense of "creating a universe from nothing" in the ordinary language sense of that term.

 

[RUTA]

Could you describe a person throwing a ball into the air and catching it on the surface of the Earth with GR using geometry and worldlines?

https://physics.stackexchange.com/q...-difference-between-clocks-in-a-gravity-field

Just read the Feynman quote for a conceptual explanation. The subsequent derivation of extremum time giving the same result as a geodesic equation follows by definition of the geodesic equation, so it’s circular. But, it does make explicit how the geodesic formalism applies to the Schwarzschild metric if you’re interested.

 

[RUTA]

[Moderator's note: split off from another thread since this is a separate topic.]

I am confused about closed timelike curves. To say that they are possible using an exact solution to the EFEs, like the Godel metric, and that they actually correspond to something in reality is where my confusion lies. What exactly does an exact solution to the EFEs tell us? Aren't they constrained by the distribution of matter density and observational facts? Does GR apply to what is possible outside of our spacetime?

Here is an Insight I wrote about CTCs https://www.physicsforums.com/insig...-4-general-relativity-closed-timelike-curves/ It may answer some of your questions.

 

[laymanB]

Here is an Insight I wrote about CTCs https://www.physicsforums.com/insig...-4-general-relativity-closed-timelike-curves/ It may answer some of your questions.

Thanks Mark. I will try to read through the rest of your Blockworld Insight series tomorrow. I do believe I am a NS inquisitor.

 

[Mordred]

How particles arise from a field takes a considerable amount of time to learn. QFT is really the tool that describes this process. The rudimentary level of understanding under QFT involve the creation and annihilation operators from some arbitrary field ground potential. This ground potential isn't necessarily zero but is set to zero as a baseline for symmetry/ assymetry relations.
One does not require treating this as VP but rather arising from anistropy variations at a certain locality more specifically amplitude variations. In essence the particle number density will increase at higher amplitudes.

Under QFT the creation and annihilation operators have the field itself as the priori. However a field is an abstract device that maps a function at every coordinate infinitisimal.
One of the paradigm shifts is to treat particles as field excitations.

Conservation of energy on a universe scale is another debate but under GR isn't considered applicable. However there is numerous laws involving particle creation and decays detailed under the Eightfold way. Another topic that takes a considerable study but a list includes conservation of charge, color, flavor, isospin, energy/momentum, parity etc.

The others have already answered the closed timelike curves. I am simply addressing the issue you raised in your first post on creation.
(I don't want to distract this thread from the topic, but felt it was worth mentioning in regards to your opening post)

 

[laymanB]

I am simply addressing the issue you raised in your first post on creation.

Thanks for the reply. I will maybe open a separate thread on particle creation in QM so as not to make this thread any more fragmented.

 

[RUTA]

I’ve had students solve some problems using the Schwarzschild metric in my GR course over the years, but I’ve never done this one — simply throwing a ball up into the air from the surface of Earth. Turns out it’s tougher than the others, but let me outline it here, since the OP asked about this. Please point out any errors, so I can fix them before making this an Insight.
Here is the Schwarzschild metric in units of $G = c = 1$:
$$ ds^2 = -\left(1 - \frac{2M}{r}\right)dt^2 + \left(1 - \frac{2M}{r}\right)^{-1}dr^2 + r^2d\Omega^2 \label{metric} $$
$d\Omega$ is the solid angle, we won’t be using the angular coordinates since our motion is strictly radial. Here are the two geodesic equations of relevance:
$$ \frac{d^2r}{dp^2} + \Gamma^{r}_{tt}\left(\frac{dt}{dp}\right)^2 + \Gamma^{r}_{rr}\left(\frac{dr}{dp}\right)^2 = 0 \label{RgdscEq} $$
and
$$ \frac{d^2t}{dp^2} + 2\Gamma^{t}_{rt}\left(\frac{dt}{dp}\right) \left(\frac{dr}{dp}\right) = 0 \label{TgdscEq} $$
where $p$ is proper time along the geodesic. We have:
$$ \Gamma^{r}_{tt} = \frac{M}{r^2} \left(1 - \frac{2M}{r}\right) \label{Christoffelrtt} $$
$$ \Gamma^{r}_{rr} = -\frac{M}{r^2} \left(1 - \frac{2M}{r}\right)^{-1} \label{Christoffelrrr} $$
$$ \Gamma^{t}_{rt} = \frac{M}{r^2} \left(1 - \frac{2M}{r}\right)^{-1} \label{Christoffelttr} $$
This gives:
$$ \frac{d}{dp}\left(\left(1 - \frac{2M}{r}\right)\frac{dt}{dp}\right) = 0 \label{TgdscEq2} $$
for our second geodesic equation, which means:
$$ \left(1 - \frac{2M}{r}\right)\frac{dt}{dp} = B \label{TgdscEq3} $$
with $B$ a constant. At $r = \infty$ and $v = 0$ the metric tells us that $dp = dt$, since $ds^2 = -dp^2$ along the geodesic, so the second geodesic equations tells us $B = 1$ when our object is launched with escape velocity $v = v_e$. If we get to $r = \infty$ with $v > 0$, then our metric tells us
$$ \frac{dp}{\sqrt{1-\frac{v^2}{c^2}}} = dt \label{SReqn} $$
(I restored $c$ here) which tells us $\frac{dt}{dp} > 1$ so our second geodesic equation says $B > 1$ when the object is launched with $v > v_e$. Therefore, we assume $B < 1$ when the object is launched with $v < v_e$. We’ll need this info on $B$ when we get to our result.
Using this result and putting our other two Christoffel symbols into our first geodesic equation gives
$$ \left(1 - \frac{2M}{r}\right)\frac{d^2r}{dp^2} + \frac{M}{r^2}\left(B^2 - \left(\frac{dr}{dp}\right)^2 \right) = 0 \label{RgdscEq2} $$
Now I will start making approximations to show this leads to the Newtonian conservation of energy equation:
$$ \frac{1}{2}v^2 - \frac{GM}{r} = \frac{E}{m} \label{Newton} $$
where $\frac{E}{m}$ is the (conserved) total energy per unit mass of the launched object (I restored $G$). Start by assuming $\frac{d^2r}{dp^2} = -g = -\frac{GM}{R^2}$ where $R$ is the radius of Earth and M is the mass of Earth, i.e., $g = 9.8 m/s^2$ per usual. Next let $r = R+y$ so that $\frac{dr}{dp} = \frac{dy}{dp} = v$, i.e., $y$ is the height above Earth’s surface of our projectile; we will assume $y$ is small compared to $R$. Now our first geodesic equation gives us:
$$ -\left(1 - \frac{2GMR}{c^2R^2\left(1+y/R\right)}\right)g + \frac{GM}{R^2\left(1+y/R\right)^2}\left(B^2 - \frac{v^2}{c^2} \right) = 0 \label{RgdscEq3} $$
where I have restored $G$ and $c$. Expanding:
$$\left(1 + y/R \right)^{-1} \approx 1 – y/R \label{approx1} $$
and
$$ \left(1 + y/R \right)^{-2} \approx 1 – 2y/R \label{approx2} $$
and putting these into our first geodesic equation and rearranging we obtain:
$$\left(\frac{1}{2} - \frac{y}{R}\right)v^2 + gy = \left(\frac{1}{2} - \frac{y}{R}\right)B^2c^2 - \frac{c^2}{2} + gR \label{RgdscEq4} $$
Since $\frac{y}{R} \ll 1$, $\left(\frac{1}{2} - \frac{y}{R}\right) \approx \frac{1}{2}$ and our first geodesic equation is now:
$$\frac{1}{2}v^2 + gy = \frac{\left(B^2 - 1\right)c^2}{2} + gR \label{RgdscEq5} $$
The $gy$ on the LHS and $gR$ on the RHS come from $-\frac{GM}{R\left(1+y/R\right)}$ expanded on the LHS of the Newtonian conservation of energy equation then using $g = \frac{GM}{R^2}$. That means $\frac{\left(B^2 - 1\right)c^2}{2}$ is our total conserved energy per unit mass $\frac{E}{m}$ from the RHS. Notice that for $v = v_e$ at launch we have $B = 1$ and the RHS of our first geodesic equation is just $gR$ as expected (total energy equals zero). If $v > v_e$ at launch, $B > 1$ and the RHS of our first geodesic equation is a little larger than $gR$ (total energy is positive). If $v < v_e$ at launch, $B < 1$ and the RHS is a little smaller than $gR$ (total energy is negative).
That concludes my outline of how the geodesic equations for the Schwarzschild metric give rise to standard Newtonian mechanics for an object launched slowly upwards near the surface of Earth. I’ll provide the details in an Insight along with other problems one can solve with the Schwarzschild metric at a later date.

 

[laymanB]

That concludes my outline of how the geodesic equations for the Schwarzschild metric give rise to standard Newtonian mechanics for an object launched slowly upwards near the surface of Earth.

Above my head right now, but I’m sure others can benefit from your considerable effort. Thanks again.

 

[PeterDonis]

That concludes my outline of how the geodesic equations for the Schwarzschild metric give rise to standard Newtonian mechanics for an object launched slowly upwards near the surface of Earth. I’ll provide the details in an Insight along with other problems one can solve with the Schwarzschild metric at a later date.

As @laymanB implies, this is probably too heavyweight for a "B" level thread. But it does look like excellent material for an Insight (or a series of them).

 

[RUTA]

As @laymanB implies, this is probably too heavyweight for a "B" level thread. But it does look like excellent material for an Insight (or a series of them).

There are several Insights on the Schwarzschild metric and problems it can solve (a series written by you, for example). I’ll look through them carefully and add the problems I’ve solved that don’t appear :-)

 

[laymanB]

What kinds of physical phenomena would we expect to see in a spacetime containing closed timelike curves?

 

[JMz]

From Carroll's article: http://www.preposterousuniverse.com/blog/2007/04/27/how-did-the-universe-start/

"Our argument shows that null and timelike geodesics are, in general, past-incomplete in inflationary models, whether or not energy conditions hold, provided only that the averaged expansion condition Hav > 0 holds along these past-directed geodesics."

I don't know enough to comment on the validity of the theorems, but they seem to show that you are left with a boundary somewhere.

I think a better phrasing of this conclusion is that CTCs are inconsistent with inflation.