Reframing and further acceleration

[brainstorm]

I am once again confused by the meaning of a material object not being able to exceed the speed of light in any frame. If the Earth is moving at 0.9C relative to some other galaxy in the universe, then would it be possible that a vessel had traveled between that galaxy and Earth? If so, wouldn't that vessel have had to accelerate to 0.9C to reach Earth? Now, if that vessel wanted to proceed past Earth to another destination, couldn't it once again accelerate to 0.9C? If not, why not, since it would reach Earth at a relatively low speed relative to Earth and could even go into orbit like any other object? In other words, doesn't any moving vessel reach a type of rest relative to itself and other objects in its frame such that acceleration to near-C velocity is repeatedly possible, even if such high velocity has already been achieved in a prior frame of motion?

 

[jtbell]

I If the Earth is moving at 0.9C relative to some other galaxy in the universe, then would it be possible that a vessel had traveled between that galaxy and Earth? If so, wouldn't that vessel have had to accelerate to 0.9C to reach Earth? Now, if that vessel wanted to proceed past Earth to another destination, couldn't it once again accelerate to 0.9C?

Yes, but this does not mean that the vessel now has a velocity of 1.8c with respect to the galaxy that it started from.

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/einvel.html

 

[brainstorm]

Yes, but this does not mean that the vessel now has a velocity of 1.8c with respect to the galaxy that it started from.

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/einvel.html

So, once it accelerates to 0.11C relative to Earth, it does not begin to surpass light emitted from its point of origin, which it previously trailed behind?

 

[Doc Al]

So, once it accelerates to 0.11C relative to Earth, it does not begin to surpass light emitted from its point of origin, which it previously trailed behind?

It does not. (What's special about 0.11c?)

Ah... you might be thinking that the speed relative to the galaxy is now 0.9c + 0.11c = 1.01c.

No, you have to compute the new velocity relativistically. The new speed ends up being only about 0.92c.

 

[JesseM]

So, once it accelerates to 0.11C relative to Earth, it does not begin to surpass light emitted from its point of origin, which it previously trailed behind?

In the frame where Earth is already traveling at 0.9c, it's impossible for the ship to be moving at 0.11c relative to the Earth (i.e. it's impossible for the speed at which the distance between the ship and the Earth is changing in this frame--the "closing speed"--to exceed 0.1c if the ship is moving in the same direction as the Earth). In the Earth's rest frame, the ship can move at 0.11c, but then according to the velocity addition formula its speed in the first frame is not 0.9c + 0.11c but rather (0.9c + 0.11c)/(1 + 0.9*0.11) = 1.01c/1.099 = 0.919c.

If it seems confusing that velocity addition works different in relativity than in classical physics, remember that in relativity each frame defines "speed" in terms of distance/time as measured by rulers and clocks at rest in that frame, but that rulers at rest in one frame appear length-contracted in another frame, while clocks at rest and synchronized in one frame appear time dilated and out-of-sync in another frame.

 

[brainstorm]

Thanks for your responses, Doc Al and Jesse M. I see that you both calculated 0.92C regarding the acceleration of the vessel from 0.9C another 0.11C. Now, can you take this back to the concrete level of the scenario I posted. Recall that the distant galaxy was expanding away from Earth at 0.9C, so by the time the ship arrived at Earth doing 0.9C relative to the point of departure, it was standing still relative to Earth. Let's just assume it takes a break and actually goes into orbit around Earth. At that point it behaves the same as anything else in orbit around Earth, right? Ok, now it decides to accelerate in its original direction from Earth to a velocity of, say, 0.5C. When someone inside looks back at its original point of departure, does it see anything at all since it is going 0.5C relative to Earth and Earth was expanding away from its original galaxy at 0.9C? Did the light waves traveling away from that galaxy fade away at the point where universal expansion exceeded C? Or are they still present, just red-shifted to a very low frequency? Sorry for all the confusion, but I don't see how accelerating from Earth to 0.11C when Earth is moving away from the galaxy at 0.9C results in a total velocity of 0.92C.

 

[jtbell]

JesseM showed you how he calculated it. It's the standard relativistic "velocity addition" formula, which can be derived from the postulates of special relativity.

 

[brainstorm]

JesseM showed you how he calculated it. It's the standard relativistic "velocity addition" formula, which can be derived from the postulates of special relativity.

I know that. I'm just trying to make sense out of how it works out in practice and what the empirical implications would be if actually observed.

 

[Bussani]

brainstorm,

I'm still no expert on it all myself, and sometimes I need to figure out a way of looking at it that makes sense just to me, or all the numbers just won't mean anything to me. If that's what you're looking for, maybe this will help; once I wrapped my head around this it made everything else somewhat more natural.

The crucial fact of it all is there's never a point where you can no longer accelerate. The Principle of Relativity tells us that no matter what velocity you're traveling at, the laws of physics will remain the same for you. Time will appear to move normally for you, you'll be able to accelerate in any direction and feel the same g-forces from it that you should at any speed, and light in a vacuum will always move at c. In other words, if you don't have something else to compare your movement to, there's no way to tell you're moving at all. This also means that you can keep accelerating constantly (so long as you have the fuel)--but you'll never hit the speed of light, of course.

The closer to c you get, the harder it is to accelerate. The acceleration still feels the same for the person doing the acceleration, but to an outside observer it's getting infinitely smaller and smaller so they never reach the speed of light. Time dilation stops you from being able to stand up on a train moving "near" the speed of light and run to the front, thus surpassing it, for instance. Of course, if you have two things both moving near the speed of light together, and nothing else around to use as a reference point, these two things could only conclude that they were both standing still together.

I think that's at the heart of it. If you had two spaceships in the middle of a completely empty universe, stationary to each other, there would be no way to say exactly how fast they were moving. It would be pointless to think about even. But let's say--just for the sake of this--that we magically know that each ship is flying through this empty space at 0.9c relative to some invisible, stationary observer or something. One ship could then still accelerate away from the other and, relative to that other one, travel away at 0.9c. But to our invisible observer, the accelerating ship would never reach or surpass c. They'd see one ship slowly, still below the speed of light, accelerate from the other, and they'd see the people on both ships moving slowly and taking longer to react to the movement. In fact, once the accelerating ship reaches 0.9c relative to the other ship, the people on the one which accelerated should be experiencing time dilation relative to other just as if they weren't moving at all; but to make it even weirder, both ships are experiencing slower time compared to our invisible, stationary observer.

If we apply it all to your original question, the galaxy that your ship started in would have watched your ship's acceleration get smaller and smaller as it approached c and would have eventually seen it reach Earth. If it could still see that far, it would see you later break orbit and move away, past Earth, even more slowly than that. By the time all of this happened, people back in your ship's home galaxy would have experienced a lot of time and would be a lot older than you in your ship.

Hopefully I've explained all of that correctly. I'm sure someone smarter will tell us if I got anything wrong. I should also note that it's all ignoring the fact that a galaxy could be receding from us at greater than c, due to space expanding or whichever explanation you prefer, since you specified that it was only at 0.9c.

 

[nutgeb]

If the .9c relative velocity of the galaxy and Earth in the example is a comoving recession velocity, then the posts in this thread are way off track.

If you assume for the purposes of the example that the Earth and the other galaxy are comoving, and the relative recession velocity is .9c then the other galaxy's proper distance is 90% of the Hubble Radius from earth, which currently is about 13.8Gly. That's a looooooong ways away. The journey would require many billions of years even if the rocket's peculiar velocity closely approached c.

It is incorrect to use the SR addition of velocity formula at large cosmological distances, particularly in a universe with gravity, such as our own. SR is used in the flat, static spacetime of Minkowski coordinates. But in the FRW coordinates typically used for cosmological distances, you cannot use the SR velocity addition formula to add a comoving recession velocity to a peculiar velocity.

In FRW coordinates, the question of what the minimum speed a rocket would need to start out with in order to reach Earth (in less than an infinite trip duration) is a complex question. It depends on the cosmological model -- whether the universe's mass/energy density is at critical density, and whether the expansion rate is dominated by radiation, matter, or Dark Energy during the relevant time period.

Also keep in mind that the Hubble recession velocity is proportional to proper distance, (H*D) where H is the current Hubble rate and D is the proper distance. Therefore, the rocket may want (or need) to continue to accelerate throughout its journey, in order to catch up with and pass successive regions of higher and higher Hubble recession velocity relative to its starting point. If it doesn't accelerate, its momentum will decrease at the rate of 1/a, where a is the then-current scale factor of the universe.

In any event, as we have been discussing in other threads here, FRW comoving recession velocities can far exceed c at great distances, so a rocket moving at .1c away from a distant galaxy (and located near that distant galaxy) that has a comoving recession velocity of, say, 5c will have a total velocity of 5.1c relative to earth. Because we are working in FRW coordinates, that does not violate SR. The rocket's peculiar velocity (its local velocity relative to its local Hubble flow) can never exceed c, and therefore the rocket could never catch up with, or even keep pace with, a photon that is ahead of it and moving the same direction. A photon located in the same FRW local reference frame is always is faster than the rocket.

 

[brainstorm]

I guess the way I should be thinking about it is the following: the vessel accelerates in any frame relative to light in that frame traveling at C. Therefore, even when it reaches 0.9C, light emitted from its point of departure is still moving at C. That light may be red-shifted in its frequency due to the relative motion between sending and receiving points, but the waves/photons are still moving at C. So does it really make sense to measure distance relative to light? It seems that you could accelerate through an endless number of frames and always remain below the C. yet relative to subsequent points in successive frames, you are traveling extremely fast. I guess all travel is relative to the rate of expansion between the departure and arrival points, though. I guess the question, then, is how much can you red-shift light by accelerating away from its source and how quickly can you move between stars according to clocks within the moving vessel?

 

[JesseM]

Thanks for your responses, Doc Al and Jesse M. I see that you both calculated 0.92C regarding the acceleration of the vessel from 0.9C another 0.11C. Now, can you take this back to the concrete level of the scenario I posted. Recall that the distant galaxy was expanding away from Earth at 0.9C, so by the time the ship arrived at Earth doing 0.9C relative to the point of departure, it was standing still relative to Earth.

How could it "arrive at Earth" if it was at rest relative to the Earth as soon as it departed the distant galaxy? If it was at rest relative to the Earth when it was still in the neighborhood of the galaxy, then the distance between it and the Earth never would have shrunk.

Also, if you want to discuss a problem in the context of special relativity, you can't assume space is expanding, since that only happens in general relativity where spacetime can be curved. It's much easier if we assume the laws of special relativity apply and the distant galaxy just happens to have a speed of 0.9c relative to the Earth in uncurved spacetime, rather than assuming space is actually expanding between them--is that OK for the purpose of discussion? In studying relativity it's best to "learn to walk before you run", to get some idea of the basics of special relativity before you try to understand how things work in general relativity.

Let's just assume it takes a break and actually goes into orbit around Earth. At that point it behaves the same as anything else in orbit around Earth, right? Ok, now it decides to accelerate in its original direction from Earth to a velocity of, say, 0.5C.

Remember, velocities can only be stated relative to a particular frame of reference. Do you mean that the ship is traveling at 0.5c in the rest frame of the Earth?

When someone inside looks back at its original point of departure, does it see anything at all since it is going 0.5C relative to Earth and Earth was expanding away from its original galaxy at 0.9C?

In the Earth's frame if the ship is moving at 0.5c, then any light signal from the distant galaxy will be traveling at 1c, so no matter how far away the galaxy is when it emits a given signal, the signal will eventually be able to catch up with the ship (again I am assuming we are dealing with ordinary flat spacetime rather than expanding space, since expanding space can mean that certain light signals will never reach certain objects). Meanwhile in the galaxy's frame, the ship is only traveling at (0.9c + 0.5c)/(1 + 0.5*0.9) = 1.4c/1.45 = 0.9655c, so it also makes sense that any signal from the galaxy will eventually catch up with the ship when we analyze things from the perspective of this frame.

Sorry for all the confusion, but I don't see how accelerating from Earth to 0.11C when Earth is moving away from the galaxy at 0.9C results in a total velocity of 0.92C.

Did you read my point about how each frame defines "speed" in terms of distance/time as measured by rulers and clocks at rest in that frame, and rulers at rest in one frame will be shrunk in other frames, while synchronized clocks at rest in one frame will be running slow and out-of-sync in other frames? For a numerical example, you might take a look at this post where I showed how two observers using their own clocks will each measure the speed of the same light beam as 1c, because of the way rulers shrink and clocks become time-dilated and out-of-sync when viewed in a frame where they're in motion (so in spite of the fact that one observer measures the light beam to move at 1c relative to himself, and the fact that the second observer measures the first to move at 0.6c relative to himself, that doesn't imply that the second observer measures the light beam to be moving at 1.6c relative to himself)

 

[brainstorm]

How could it "arrive at Earth" if it was at rest relative to the Earth as soon as it departed the distant galaxy? If it was at rest relative to the Earth when it was still in the neighborhood of the galaxy, then the distance between it and the Earth never would have shrunk.

Also, if you want to discuss a problem in the context of special relativity, you can't assume space is expanding, since that only happens in general relativity where spacetime can be curved. It's much easier if we assume the laws of special relativity apply and the distant galaxy just happens to have a speed of 0.9c relative to the Earth in uncurved spacetime, rather than assuming space is actually expanding between them--is that OK for the purpose of discussion? In studying relativity it's best to "learn to walk before you run", to get some idea of the basics of special relativity before you try to understand how things work in general relativity.

I don't see the difference between saying the Earth and the galaxy are moving away from each other at 0.9C and saying that the universe is expanding such that it appears so. Space is irrelevant to me except as gravitational/energy relations between objects. Having said that, I see your point that by the time you reach 0.9C in the distant galaxy, you are just standing still relative to Earth, so at that point you have to start accelerating relative to Earth to approach it.

This raises new issues regarding the light emitted from both the galaxy and Earth, however: Once you have accelerated to 0.9C, the light from the departure galaxy has red-shifted while the light from Earth has blue-shifted, correct? So my question becomes what the difference is between being very far from Earth but standing still relative to it, or being very far and moving away from it at 0.9C. Just the frequency of light-emissions from it, right?

 

[nutgeb]

brainstorm, you're just going to confuse yourself if you try to pose your question as a GR cosmology question and an SR inertial frame question at the same time. You have to ask the question one way or the other. Part of walking before you run is being reasonably specific in how you frame your question. It's your choice.

The original example of a galaxy having a recession velocity of .9c would be a very nonrealistic hypothetical example if you are talking about a flat, static Minkowski frame. There are no galaxies near enough to Earth (such that the effect of spacetime curvature is minimal) that have anywhere near a .9c peculiar velocity, and if they are much farther away they cannot possibly be considered to be in the same inertial frame as earth. But if that is the hypothetical you are asking about, then you should use Minkowski coordiantes and apply the SR addition of velocities formula.

If however you are asking about a more realistic example where .9c represents a distant galaxy's comoving recession velocity, then you should apply FRW coordinates, in which case you cannot apply the SR addition of velocities formula. Hubble's Law is 'built into' the FRW coordinate system, and recession velocities can exceed c without bound. The current proper distance from Earth to a galaxy with a .9c comoving recession velocity would be approximately 12.4 Gly. If its speed is substantially less than c, a rocket might never be able to complete the journey even in an infinite amount of time, due to the acceleration of the expansion rate due to Dark Energy.

I think it is misleading to suggest that the 'expanding space' paradigm, or for that matter the applicability of FRW coordinates, applies only in a universe with gravity (spacetime curvature). FRW coordinates are applied all the time to model 'open' infinite universes with vanishingly small spacetime curvature, of which the Milne model is the best known example. When the Milne universe is modeled in FRW coordinates, it is typical to apply either the 'expanding space' paradigm, or alternatively the 'kinematic' paradigm, as the explanation for why and how Hubble's Law works in FRW coordinates.

Alternatively, SR Minkowski coordinates can be used to analyze the Milne model, and they generate exactly the same observations, such as redshift. But as a result of the transformation to Minkowski from FRW coordinates, recession velocities never exceed c, and radial distances at high recession velocities are dramatically Lorentz-contracted, relative to FRW proper distance coordinates. See http://world.std.com/~mmcirvin/milne.html" for some accessible explanations. It's good stuff to learn about.

 

[JesseM]

I don't see the difference between saying the Earth and the galaxy are moving away from each other at 0.9C and saying that the universe is expanding such that it appears so.

Because "expanding space" requires curved spacetime, and the laws of physics don't obey the same equations in any coordinate system covering a large region of curved spacetime that they do in an inertial coordinate system in the flat spacetime of SR. For example, in coordinate systems covering curved spacetime you can't assume light travels at the same speed c everywhere, or that all massive objects travel slower than light (as 'speed' is defined in that coordinate system)

This raises new issues regarding the light emitted from both the galaxy and Earth, however: Once you have accelerated to 0.9C, the light from the departure galaxy has red-shifted while the light from Earth has blue-shifted, correct?

Light from Earth has blue-shifted as compared to the frequency it was when you were at rest relative to the galaxy, but if you're at rest relative to the Earth it's neither red-shifted nor blue-shifted as compared with the frequency measured by Earthlings (again assuming we are talking about SR rather than GR).

So my question becomes what the difference is between being very far from Earth but standing still relative to it, or being very far and moving away from it at 0.9C. Just the frequency of light-emissions from it, right?

In terms of what you see, the frequency is different depending on your velocity relative to it, yes.

 

[brainstorm]

brainstorm, you're just going to confuse yourself if you try to pose your question as a GR cosmology question and an SR inertial frame question at the same time. You have to ask the question one way or the other. Part of walking before you run is being reasonably specific in how you frame your question. It's your choice.

I realize that many people like to work with paradigm-specific axiomatism but I don't, mainly because it is not my job the way it is for an academic physicist. Personally, I prefer to reason about an issue directly and only compare different paradigms where the necessity arises because of a "fork in the road." What I think would be interesting for lay people like me is to address the issues in the way Bussani did (I'd put the post # but it doesn't show up when I scroll down in this screen). In other words, I would like to simply get a concrete idea of what is happening to an object that is accelerating vis-a-vis the light around it and the distance between the object and other objects it is approaching or moving away from. I think I have come to a clear understanding of how light beams redshift as you accelerate parallel to them. I thought I also understood correctly that distant stars move away from each other, either due to universal expansion or simply because they are moving away from each other due to their momentum. Like I said, I don't get the difference between these, so all that really makes sense to me is that they are moving away from each other at a certain speed. Then, the big final question for me is how do you know how fast you're traveling except by measuring the redshift or blueshift of light from your points of departure or destination? Likewise, I'm wondering if there's a limit to the amount of redshift or blueshift you can experience relative to a particular beam of light. Is it possible, for instance, to be moving fast enough toward Earth that radio wave emissions would appear as gamma rays? If so, how would visible light appear? As something with a higher frequency than gamma radiation?