Can Accelerate at Constant g in Own FoR?

[S Holtom]

TL;DR Summary

Can you accelerate forever, at constant g, in your own frame of reference?

( This question came up in the context of flat earth. Sorry for that, but hopefully the question itself is not dumbtarded. )

Basically one "model" for gravity that some FErs subscribe to, is that the Earth disc is accelerating upwards at 9.8 ms-2. And one debunk of this proposition was that the disc Earth would reach c in a mere 353 days, and then would need to exceed c which is impossible.

But, ISTM, this is looking at it classically (but with a speed limit bolted on).

But what about looking at it relativistically? If we have infinite energy we can indeed accelerate forever, but, from the point of an intertial observer, this acceleration tends to zero. But how about from our own frame of reference? Can we accelerate at what would locally feel like 9.8ms-2 forever?

 

[Dale]

Yes, this is the idea behind the relativistic rocket equations

https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

 

[PeroK]

Summary:: Can you accelerate forever, at constant g, in your own frame of reference?

But what about looking at it relativistically? Can we accelerate at what would locally feel like 9.8ms-2 forever?

An object on the Earth's surface has constant proper acceleration of $9.8 \ m/s^2$ upwards. That's a possibility in curved spacetime.

If spacetime is flat, then your velocity relative to the rest of the universe would be increasing. That idea won't work.

 

[PeterDonis]

If spacetime is flat, then your velocity relative to the rest of the universe would be increasing.

If spacetime is flat, there is no "rest of the universe"; the spacetime has no stress-energy in it anywhere. It is perfectly possible (assuming you have some inexhaustible source of thrust) to accelerate at a constant proper acceleration forever in flat spacetime.

 

[ergospherical]

If spacetime is flat, there is no "rest of the universe"; the spacetime has no stress-energy in it anywhere. It is perfectly possible (assuming you have some inexhaustible source of thrust) to accelerate at a constant proper acceleration forever in flat spacetime.

What about the exhaust gases, or equivalent?
(My point being that any method of propulsion must satisfy momentum conservation).

 

[PeroK]

It is perfectly possible (assuming you have some inexhaustible source of thrust) to accelerate at a constant proper acceleration forever in flat spacetime.

The two problems with that hypothesis are the assumption of inexhaustible energy and the lack of red/blue shift for everything else. Not to mention the lack of relative motion.

 

[PeterDonis]

What about the exhaust gases, or equivalent?
(My point being that any method of propulsion must satisfy momentum conservation).

If we are not considering the rocket and its exhaust gases as test objects, i.e., if they have nonzero stress-energy, then we aren't in flat spacetime to begin with. But @PeroK, whom I was responding to, started with the assumption that spacetime was flat. That requires us to consider the rocket and its exhaust gases as test objects, with zero (or at least negligible) stress-energy.

 

[PeterDonis]

The two problems with that hypothesis are the assumption of inexhaustible energy

If we're in flat spacetime, which was your hypothesis in the part of your post that I responded to, then, as I responded to @ergospherical just now, then we have to consider the rocket and its fuel, exhaust, etc., as test objects, with zero (or at least negligible) stress-energy. That means there is no issue with "inexhaustible energy" in this highly idealized scenario.

and the lack of red/blue shift for everything else. Not to mention the lack of relative motion.

What "everything else"? Again, flat spacetime was your hypothesis in the part of your post that I responded to. Under that hypothesis there is no "everything else". If there is "everything else", then spacetime is not flat.

In other words, the part of your post that I responded to...

If spacetime is flat, then your velocity relative to the rest of the universe would be increasing.

...is internally inconsistent: you can't hypothesize both that spacetime is flat and that there is a "rest of the universe" for your velocity to be increasing relative to. That was my point.

 

[PeroK]

If we're in flat spacetime, which was your hypothesis in the part of your post that I responded to, then, as I responded to @ergospherical just now, then we have to consider the rocket and its fuel, exhaust, etc., as test objects, with zero (or at least negligible) stress-energy. That means there is no issue with "inexhaustible energy" in this highly idealized scenario.What "everything else"? Again, flat spacetime was your hypothesis in the part of your post that I responded to. Under that hypothesis there is no "everything else". If there is "everything else", then spacetime is not flat.

In other words, the part of your post that I responded to...

...is internally inconsistent: you can't hypothesize both that spacetime is flat and that there is a "rest of the universe" for your velocity to be increasing relative to. That was my point.

Spacetime is flat in SR - without a law of gravity. That's not internally inconsistent. That was one of the OP's hypotheses that he wanted an opinion on, as I understand it.

 

[PeroK]

But what about looking at it relativistically? If we have infinite energy we can indeed accelerate forever, but, from the point of an intertial observer, this acceleration tends to zero. But how about from our own frame of reference? Can we accelerate at what would locally feel like 9.8ms-2 forever?

This is what I addressed. The hypothesis that there is no gravity, only flat spacetime and (unexplained) proper acceleration.

 

[PeterDonis]

Basically one "model" for gravity that some FErs subscribe to, is that the Earth disc is accelerating upwards at 9.8 ms-2. And one debunk of this proposition was that the disc Earth would reach c in a mere 353 days, and then would need to exceed c which is impossible.

But, ISTM, this is looking at it classically (but with a speed limit bolted on).

But what about looking at it relativistically? If we have infinite energy we can indeed accelerate forever, but, from the point of an intertial observer, this acceleration tends to zero. But how about from our own frame of reference? Can we accelerate at what would locally feel like 9.8ms-2 forever?

Debunking flat earther models is the sort of thing that would be listed in the Hitchhiker's Guide to the Galaxy under "recreational impossibilities", since FErs are impervious to actual evidence and reasoning. However, for anyone who is not impervious to actual evidence and reasoning, if you are going to accept relativity, you have to accept that relativity says that a "flat disc" Earth is not stable under its own gravity. Nor can you dodge this by saying that the "gravity" is due to acceleration "upward" of the flat disc, which itself has negligible mass, because the energy that would be needed to power that acceleration is itself a source of gravity and would make a flat disc Earth unstable.

 

[PeterDonis]

Spacetime is flat in SR - without a law of gravity. That's not internally inconsistent.

Hm. So the hypothesis is that everything in the universe is a "test object". That would imply that our cosmological model for the universe as a whole would have to be the Milne model (galaxies flying apart in flat spacetime from a common starting point far in the past). But we already know that model is inconsistent with observation. (Of course, as I noted in post #11 just now, FErs are impervious to the whole "inconsistent with observation" thing anyway, so I don't know that anything we've said in this thread would change a FEr's mind.)

 

[S Holtom]

Thanks for the responses so far.

The FE thing was just for context / interest, so let's just remove it, and I'll try to rephrase:

Under general relativity, is it possible to experience a constant g due to acceleration (rather than mass) indefinitely?
Disregarding where the energy for acceleration is coming from, or energy issues in general (eg I'd guess at a certain point the energy density would be high enough for any such object to become a black hole, but let's ignore this for now)

 

[PeroK]

Under general relativity, is it possible to experience a constant g due to acceleration (rather than mass) indefinitely?

Yes. There's no limit as long as you have the source of energy to generate the constant acceleration.

Disregarding where the energy for acceleration is coming from, or energy issues in general (eg I'd guess at a certain point the energy density would be high enough for any such object to become a black hole

No. You cannot turn into a black hole. All motion is relative in this sense and acceleration does not cause a physical difference in an object.

You are alluding to relativistic mass, I imagine, which is a not a physical concept and is particularly misleading vis-a-vis gravity.

 

[PeterDonis]

Under general relativity, is it possible to experience a constant g due to acceleration (rather than mass) indefinitely?
Disregarding where the energy for acceleration is coming from, or energy issues in general

You can't disregard where the energy is coming from, or energy issues in general, if you are asking the question in the context of GR. So the question as you pose it here is inconsistent.

There's no limit as long as you have the source of energy to generate the constant acceleration.

In GR, you can't have enough energy to generate constant acceleration indefinitely; that would require an infinite amount of energy, which would cause an infinite amount of spacetime curvature, which is impossible. The question as it was posed this time said "under general relativity", and in GR you can't ignore the fact that energy causes spacetime curvature.

 

[jbriggs444]

In GR, you can't have enough energy to generate constant acceleration indefinitely; that would require an infinite amount of energy, which would cause an infinite amount of spacetime curvature, which is impossible

As long as you are willing to shrink your payload indefinitely (i.e. ignoring any quantum complications and assuming a continuum model), you can thrust forever, becoming smaller and smaller as you go.

This avoids any problem with infinite energy or infinite energy density.

 

[PeterDonis]

Thread closed for moderation.

[Edit: Thread reopened after cleanup and some offline discussion.]

 

[PeterDonis]

As long as you are willing to shrink your payload indefinitely (i.e. ignoring any quantum complications and assuming a continuum model), you can thrust forever, becoming smaller and smaller as you go.

I had objected to this originally, but based on PM conversation, I now understand the type of model you are proposing here and I agree that it is consistent. (It does raise an additional question about an alternate possibility which I'll save for a separate post.)

Using the relativistic rocket equations in the article @Dale linked to in post #2, we have the following (I am using units in which $c = 1$):

$$
v = \frac{at}{\sqrt{ 1 + a^2 t^2}}
$$

$$
\gamma = \frac{1}{\sqrt{1 - v^2}} = \sqrt{ 1 + a^2 t^2 }
$$

$$
\frac{M + m}{m} = \gamma \left( 1 + v \right)
$$

The last equation is slightly rewritten from the FAQ article. What it gives is the ratio of the total mass present at the start (when the rocket is sitting at rest on the launch pad) to the rest mass of the remaining portion of the rocket, at the time when the rocket has achieved speed $v$. Note that the FAQ article interprets this point as the end of the burn, so that $m$ is the rest mass of the "payload", i.e., rest mass that never gets converted into rocket exhaust. However, we can just as easily apply the same equation to a burn that goes on indefinitely, i.e., for the case $t \to \infty$, $v \to 1$.

To facilitate that interpretation, we will change notation slightly; instead of $M + m$, we will write $E$, since this quantity is the total energy present in the scenario. (Since at the start, everything is at rest on the launch pad, $E$ is also the total rest mass that is sitting on the launch pad at the start.). Then our third equation becomes:

$$
\frac{E}{m} = \gamma \left( 1 + v \right)
$$

This now tells us that, since $\gamma \left( 1 + v \right) \to \infty$ as $v \to 1$, we must have $m \to 0$ as $v \to 1$ (since $E$ is a fixed finite number). That is what "shrinking the payload indefinitely" refers to. Now we simply need to work out the consequences.

The above equation obviously implies

$$
m = \frac{E}{\gamma \left( 1 + v \right)}
$$

We know that we must have energy conservation and momentum conservation, so we can write down two more equations as follows:

$$
E = E_r + E_e = \gamma m + E_e
$$

$$
P = 0 = P_r + P_e = \gamma m v + P_e
$$

where $E_r$ and $P_r$ are the total energy and momentum of the rocket, $E_e$ and $P_e$ are the total energy and momentum of the rocket exhaust. The above equations give us the following equations for the energies and momenta:

$$
E_r = \frac{E}{1 + v}
$$

$$
P_r = \frac{E v}{1 + v}
$$

$$
E_e = \frac{E v}{1 + v}
$$

$$
P_e = - \frac{Ev}{1 + v}
$$

These equations tell us several things. First, since we have $|E_e| = |P_e|$, the rocket exhaust is indeed photons (as it should be, since our equations are all based on the assumption of a photon rocket). Second, in the limit $v \to 1$, we have $E_r = E_e$, i.e., half of the original energy ends up in the rocket and half ends up in the exhaust. And third, in the limit $v \to 1$, we have $E_r = P_r$, which makes sense mathematically since the hyperbolic worldline of the rocket asymptotes to a null line, i.e., to the worldline of a photon.

An interesting point about this, though, is that, even though the payload "shrinks indefinitely", to zero rest mass in the limit, it does not "vanish"--the payload has nonzero energy and momentum even in the limit. This observation raises a follow-up question that I'll put in a separate post.

 

[PeterDonis]

This observation raises a follow-up question that I'll put in a separate post.

The follow-up question is simple: is there an alternate model where the payload not only "shrinks indefinitely" in the sense of $m \to 0$ in the limit, but also actually "vanishes", i.e., $E_r \to 0$ and $P_r \to 0$ in the limit? This would mean that all of the initial energy $E$ ends up in the rocket exhaust energy $E_e$, and that the total momentum $P_e$ of the rocket exhaust would have to go to zero in the limit.

It is obvious that such a model cannot be based on a photon rocket, since the solution in the previous post is the only possible one for a photon rocket. But I'm wondering if it is possible to find such a solution for a rocket whose exhaust moves on timelike worldlines, not null worldlines. For such a case, it at least seems possible heuristically that the rocket exhaust momentum could vanish in the limit, since at some point during the burn the rocket exhaust, even though it is moving to the left relative to the rocket, would be moving to the right relative to the original rest frame, and would therefore have positive instead of negative momentum in that frame.

What I have not been able to come up with is either an actual solution that has these properties, or a proof that no such solution can exist. (More precisely, that no such solution can exist which has constant proper acceleration for the rocket.)

 

[PeterDonis]

I now understand the type of model you are proposing here and I agree that it is consistent.

I should note, though, that a model of this type cannot explain constant proper acceleration indefinitely for an object like the Earth, since the Earth's rest mass cannot be reduced to arbitrarily small values.

 

[PeterDonis]

Under general relativity, is it possible to experience a constant g due to acceleration (rather than mass) indefinitely?
Disregarding where the energy for acceleration is coming from

I should note that there is a known solution in GR (i.e., a curved spacetime that takes into account the stress-energy involved) that describes this kind of scenario: the Kinnersley photon rocket. Unfortunately, I don't think Kinnersley's original paper describing this solution is online, but I believe we have had previous PF threads on it. I also don't know off the top of my head if the asymptotic properties of this solution are the same as the ones described by the flat spacetime solution in post #18.

 

[S Holtom]

All motion is relative in this sense and acceleration does not cause a physical difference in an object.

You are alluding to relativistic mass, I imagine, which is a not a physical concept and is particularly misleading vis-a-vis gravity.

No I was alluding to things like a kugelblitz. It's possible to make black holes from zero mass, all you need is a mass-energy density within a volume smaller than its schwarzschild radius. (Where, AIUI, "mass-energy" is just the sum total, so can be entirely energy). I thought if I start positing an unlimited energy resource then this kind of issue might come up.
So I was just trying to exclude this from the hypothetical.

 

[PeroK]

No I was alluding to things like a kugelblitz.

Given that the thread is entitled "can you accelerate at constant $g$ in your own frame of reference", I think you fooled everyone on that point.

 

[Ibix]

I would say that it is uncontroversial to state that you can draw a worldline (a path through spacetime) that represents eternal constant proper acceleration, and that this does not involve exceeding $c$. However, I don't think anything (except possibly the idealised cases jbriggs444 and PeterDonis were discussing) could actually follow it because anything even slightly non-idealised would run out of fuel eventually.

 

[S Holtom]

Uncontroversial perhaps, but it wasn't obvious to me as a non-physicist, so I appreciate the answer :)

 

[Ibix]

Uncontroversial perhaps, but it wasn't obvious to me as a non-physicist, so I appreciate the answer :)

It wasn't a dig. I was just drawing a distinction between the stuff I'm pretty sure all contributors will agree on and stuff that's apparently more arguable.

 

[PeterDonis]

I would say that it is uncontroversial to state that you can draw a worldline (a path through spacetime) that represents eternal constant proper acceleration, and that this does not involve exceeding $c$.

Yes.

However, I don't think anything (except possibly the idealised cases jbriggs444 and PeterDonis were discussing) could actually follow it because anything even slightly non-idealised would run out of fuel eventually.

Yes. In the idealized case where an object could have constant proper acceleration indefinitely with a finite amount of fuel, the object would have to shrink indefinitely as well. So there could not be any actual "payload" (i.e., something not eventually converted to rocket exhaust) in such a case.

 

[Ibix]

So there could not be any actual "payload" (i.e., something not eventually converted to rocket exhaust) in such a case.

Except possibly a stress-energy free spherical cow.

 

[Dale]

You guys are getting too bogged down in technical details. There is no finite limit to how long you can accelerate in your own frame of reference. And there is no need to restrict yourself to rockets that carry all of their fuel as payload.

It is fine to say that yes it is possible to accelerate forever in your frame.

 

[Filip Larsen]

It may be this discussion has wandered a bit now, but I am still stuck being puzzled that someone would think that a flat-earther would believe SR or GR to be real enough to alter their belief of a flat Earth. I mean, their argument would undoubtedly be that if SR or GR in any way contradict a flat Earth, then SR or GR must "obviously" be invalid, while the inverse probably don't mean anything to them. I think arguing with a flat-earther is more about psychology than physics. If you strongly disbelieve that "1+1 = 2" with all the normal interpretations of the symbols, then you are not going to be persuaded by arguments that, say, dip into the strangeness of the real numbers.

OK, sorry for the digression, please carry on.

 

[jbergman]

You guys are getting too bogged down in technical details. There is no finite limit to how long you can accelerate in your own frame of reference. And there is no need to restrict yourself to rockets that carry all of their fuel as payload.

It is fine to say that yes it is possible to accelerate forever in your frame.

Yup. It's even described in mtw for special relativity in chapter 6, I believe. Basically you asymptotically approach the speed of light and follow a hyperbolic path in space time.

 

[PeterDonis]

You guys are getting too bogged down in technical details.

Granted, this is a "B" level thread, but I still think it's worth pointing out the distinction between idealized models and what can actually be physically realized under what conditions.

I think @Ibix in post #24 summed things up pretty well.

 

[Dale]

Granted, this is a "B" level thread, but I still think it's worth pointing out the distinction between idealized models and what can actually be physically realized under what conditions.

I think @Ibix in post #24 summed things up pretty well.

Yes, but in post 13 the OP basically asked to focus on the idealized model, which is a perfectly reasonable request and which we have done a rather poor job of now.

 

[PeterDonis]

in post 13 the OP basically asked to focus on the idealized model

Sort of. The "idealized model" he was proposing was inconsistent. See my response in the first paragraph of post #15. Sometimes the best response to a question is to point out issues with the question rather than to just provide an answer as it is asked.