Is Faster-than-Light Travel Possible with Constant Acceleration?

[nahkaimurrao]

While SR states one can never measure a velocity greater than C, for all practical purposes one can travel 'faster than light'. Here is what I mean. . .

Lets assume I have a Spaceship with the means of constantly accelerating at 1G. The fact that this is technologically unfeasible is irrelevant to the principles involved.

We choose a destination and for our purposes assume linear travel.

I calculate that one is able to travel many light years of distance within a human lifetime.
We will base this though experiment on SR and traveling at constant acceleration of 1G.

We will acclerate for the first half of the trip and decelerate for the second half. Obviously would could just keep accelerating forever if we wanted to get there faster, but this would leave us going to fast to enjoy the destination.

I calculate the time it takes using:
Time(years) = SQRT(Distance/2*2/accel)*2

I attached a Excel file showing various times for various distances you want to travel at 1G or 2G in your POV and Earth's POV.

In summary from Earth's POV you will asymptotically accelerate toward C so for long distances it would take you roughly 1 year per light year.

But from your POV, for practical purposes, you would be able to travel:

4 light years in 3.94 years using 3.7E20 J of energy
100 light years in 19.7 years using 9.3E21 J of energy
1000 light years in 62.31 years using 9.3E22 J of energy

Given 10^22 J of energy you could feasibly travel 1000 light-years in your lifetime!(what is really happening is that the energy is shrinking space to allow you to travel that far, but for all practical purposes of travel this works)

to put the energy in perspective, a large nuclear exploxive can give 8.4E13 J of energy so we would need 10 million times this much energy to travel 4 light years in 3.94 years which is entirely possible but highly improbable.

 

[russ_watters]

Yes, you are right that due to time dilation and length contraction, you can travel seemingly long distances (measured from your pre-launch stationary frame) in a short time, without ever measuring your speed to be greater than C. If we ever get the technology to accelerate a spacecraft to a high fraction of C, we could send spaceships to distant stars and back within the lifespan of the crew. The catch, of course, is everyone who helped build the craft will be long dead when they get back.

 

[Dale]

Hi nahkaimurrao,

This is also quite well described on the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html" page. I didn't look at your excel spreadsheet, but you can check your calculations with theirs.

 

[nahkaimurrao]

Thanks
It appears my calculations are different than those, but I think I understand the concept at least.
Mostly I wanted to point out that it is possible to travel to distant star/galaxies within a reasonable time (from the traveler's POV). This doesn't seem to be understood by the mainstream public who tend to think that it would take at least 1 yr to go 1 lightyear etc.

I feel that saying the speed of light is a barrier is misleading. It is more of another dimension. When we accelerate, out frame of reference seems to disassociate with Earth's frame of reference. We are literally 'warping' space-time. The speed of light only appears to be a barrier because as you warp space you are creating a new space with it's own properties distinct from that of other spaces and within this space the speed of light is still observed.

 

[MeJennifer]

I feel that saying the speed of light is a barrier is misleading. It is more of another dimension. When we accelerate, out frame of reference seems to disassociate with Earth's frame of reference. We are literally 'warping' space-time. The speed of light only appears to be a barrier because as you warp space you are creating a new space with it's own properties distinct from that of other spaces and within this space the speed of light is still observed.

Actually I think the term 'warping' is misleading here since your example is obviously using flat spacetime.

 

[aranoff]

Instead of discussing speed of light, talk about speed of information. Gravitation also travels at the speed c. Physical objects consist of information. Information cannot travel faster than itself.

 

[Dale]

If you want to avoid the term "speed of light" the usual term is "invariant speed".

 

[aranoff]

Thanks
It appears my calculations are different than those, but I think I understand the concept at least.
Mostly I wanted to point out that it is possible to travel to distant star/galaxies within a reasonable time (from the traveler's POV). This doesn't seem to be understood by the mainstream public who tend to think that it would take at least 1 yr to go 1 lightyear etc.

I feel that saying the speed of light is a barrier is misleading. It is more of another dimension. When we accelerate, out frame of reference seems to disassociate with Earth's frame of reference. We are literally 'warping' space-time. The speed of light only appears to be a barrier because as you warp space you are creating a new space with it's own properties distinct from that of other spaces and within this space the speed of light is still observed.

It does take a year to travel one light year. For the external observer. How about him returning? Twin paradox.

 

[nahkaimurrao]

Actually I think the term 'warping' is misleading here since your example is obviously using flat spacetime.

What I mean by 'warping' is that as one person accelerates, his reference frame (both space and time) lose continuity with that of his initial reference frame (perhaps his twin who stayed at home).

It is almost like you are becoming 'distant' but by a dimension distinct from those we normally experience. It is analogous to communication over long distance, say between Earth and mars. If both people talk at the same time (sending radio waves to the other's planet) the other person appears to be delayed. But which person spoke first? Each person observes that they spoke first because the other's message was delayed by distance.

When two beings are traveling at greatly differing velocities, each appears to the other to have slowed down. But who is it that truly slowed down? It is unclear becomes space has become less linear, more 'warped' so that different observers measure different time/dimensions. Time runs differently for each observer, and space measures differently for each observe.

 

[MeJennifer]

It is unclear becomes space has become less linear, more 'warped' so that different observers measure different time/dimensions.

Again that has nothing to do with warping of space.

 

[nahkaimurrao]

Again that has nothing to do with warping of space.

I'm afraid your rebuttal leaves something to be desired. Would you please enlighten me with your understanding of the 'warping of space' and how it differs from my illustration thus far?

 

[aranoff]

Once we discuss acceleration, we must discuss general relativity, not simply special relativity. The basic equation of GR is G=T. G is geometry, "warping of space". That is, with acceleration, we cannot have "flat space".

 

[JesseM]

Once we discuss acceleration, we must discuss general relativity, not simply special relativity. The basic equation of GR is G=T. G is geometry, "warping of space". That is, with acceleration, we cannot have "flat space".

Not true, you can certainly have accelerating test particles (with negligible mass so they don't curve spacetime) in flat spacetime. See for example this page from the Usenet Physics FAQ:

It is a common misconception that Special Relativity cannot handle accelerating objects or accelerating reference frames. It is claimed that general relativity is required because special relativity only applies to inertial frames. This is not true. Special relativity treats accelerating frames differently from inertial frames but can still deal with them. Accelerating objects can be dealt with without even calling upon accelerating frames.

This error often comes up in the context of the twin paradox when people claim that it can only be resolved in general relativity because of acceleration. This is not the case.

The only sense in which special relativity is an approximation when there are accelerating bodies is that gravitational effects such as generation of gravitational waves are being ignored. But of course there are larger gravitational effects being neglected even when massive bodies are not accelerating and they are small for many applications so this is not strictly relevant. Special relativity gives a completely self-consistent description of the mechanics of accelerating bodies neglecting gravitation, just as Newtonian mechanics did.

The difference between general and special relativity is that in the general theory all frames of reference including spinning and accelerating frames are treated on an equal footing. In special relativity accelerating frames are different from inertial frames. Velocities are relative but acceleration is treated as absolute. In general relativity all motion is relative. To accommodate this change general relativity has to use curved space-time. In special relativity space-time is always flat.

In special relativity an accelerating particle has a worldline which is not straight. This is not difficult to handle. The 4-vector acceleration can be defined as the derivative with respect to proper time of the 4-velocity. It is possible to solve the equations of motion for a particle in electric and magnetic fields, for example.

Accelerating reference frames are a different matter. In GR the physical equations take the same form in any co-ordinate system. In SR they do not but it is still possible to use co-ordinate systems corresponding to accelerating or rotating frames of reference just as it is possible to solve ordinary mechanics problems in curvilinear co-ordinate systems. This is done by introducing a metric tensor. The formalism is very similar to that of many general relativity problems but it is still special relativity so long as the space-time is constrained to be flat and Minkowskian. Note that the speed of light is rarely constant in non-inertial frames and this has been known to cause confusion.

 

[granpa]

Yes, you are right that due to time dilation and length contraction, you can travel seemingly long distances (measured from your pre-launch stationary frame) in a short time, without ever measuring your speed to be greater than C.

I don't think length contraction is relevant. only the time dilation matters.

or rather, you get the benefit of one or the other but not both at the same time.

 

[Dale]

I'm afraid your rebuttal leaves something to be desired. Would you please enlighten me with your understanding of the 'warping of space' and how it differs from my illustration thus far?

I think you two are just having a miscommunication. Spacetime "warping" as it is usually used refers specifically to spacetime having a non-zero curvature which can cause inertially moving observers to accelerate wrt each other (which doesn't happen in flat spacetime even with accelerating observers). You are referring to "warping" as any change in time or distance measurements. An imprecise analogy would be MeJennifer referring to bending a metal bar and you referring to stretching it both using the same word.

 

[russ_watters]

I don't think length contraction is relevant. only the time dilation matters.

or rather, you get the benefit of one or the other but not both at the same time.

They are two parts of the same thing. The reason you can travel to, say, Alpha Centuari in less than 4.5 years (at close to the speed of light) even though it is 4.5 years away is you measure the distance to be much smaller than 4.5 LY.

Heck, it may actually be better to say that from your point of view (the traveler), it is length contraction alone that allows you to get there that fast.

 

[nahkaimurrao]

Thanks for all the replies. I admit I have not had any formal education regarding GR and only undergrad classes regarding SR. Other than that I was basically going off intuition.

Is it generally understood according to SR that a photon simultaneously exists everywhere along its path i.e. that in its POV there is no distance from beginning to end?

 

[granpa]

Heck, it may actually be better to say that from your point of view (the traveler), it is length contraction alone that allows you to get there that fast.

yes. from the point of view of the stationary observer only time dilation matters. from the point of view of the traveler, only the length contraction matters.

 

[JesseM]

Is it generally understood according to SR that a photon simultaneously exists everywhere along its path i.e. that in its POV there is no distance from beginning to end?

A photon doesn't actually have its own POV (i.e. its own inertial rest frame) in SR. All inertial frames are ones that are the rest frames of sublight objects. This has to do with the assumption that the laws of physics should work the same in all inertial frames (which wouldn't work if there was a frame where photons can be at rest), and also to do with the fact that the coordinates of each frame are supposed to be defined in terms of measurements on rulers and clocks at rest in that frame, and one can't construct rulers or clocks that move at the speed of light.

 

[nahkaimurrao]

This may be more philosophy than physics but what you said makes sense assuming what I said is true, that light exists simultaneously everywhere along its path as if there is no such thing as distance, space or time. In other words in the photons POV there is no POV because spacetime hardly exists at all.

Two distant electrons that communicate via photon are connected (via the photon's POV) by zero distance or time as if they were the same electron, this seems to support the "one electron" universe model.

 

[HallsofIvy]

PHOTONS have no "POV" and PHOTONS do not "experience" time. In any frame of reference moving at less than the speed of light relative to anything else, in other words wherever we live, photons move at "the speed of light" and are not "simultaneously everywhere".

 

[WillBlake]

There is some discussion about traveling faster than the speed of light, which is theorized to be constant. We are in between two limits: zero and the speed of light. I do not hear much discussion on traveling slower than zero. I believe this as much a topic to be explored as it is one of the two limits. If the speed of light is considered to be the starting point (zero) then it should be possible to go more slowly than the artificial limit of zero. Answering this dilemma will clarify why we can't go faster than the speed of light.

 

[Dale]

light exists simultaneously everywhere along its path as if there is no such thing as distance, space or time.

The comments by others that a photon does not have a reference frame or a point of view are all correct. However, there is one thing that you can correctly say that is somewhat related to your comment here. Specifically: the http://www.physics.fsu.edu/users/ProsperH/AST3033/relativity/Interval.htm" is 0 between any two events on a photon's worldline.

 

[JesseM]

There is some discussion about traveling faster than the speed of light, which is theorized to be constant. We are in between two limits: zero and the speed of light. I do not hear much discussion on traveling slower than zero. I believe this as much a topic to be explored as it is one of the two limits. If the speed of light is considered to be the starting point (zero) then it should be possible to go more slowly than the artificial limit of zero. Answering this dilemma will clarify why we can't go faster than the speed of light.

You can certainly have a velocity less than zero in some direction because velocity is a directional quantity, so if you ask what your velocity is to the right and you're actually moving to the left, then you're defined to have a negative velocity to the right. But speed is defined as the absolute value of velocity, and absolute value is always greater than or equal to zero by definition (for example, the absolute value of -3 is 3)

 

[WillBlake]

If zero is by definition, then so is the limit to the speed possible which is c. It still does not answer the question of why.

 

[JesseM]

If zero is by definition, then so is the limit to the speed possible which is c. It still does not answer the question of why.

There is nothing in the definition of "speed" which limits speeds to c; this limit is not a matter of definition, but rather a consequence of the laws of physics in our universe. It's perfectly possible to imagine a hypothetical universe with different laws where there is no upper limit on speed (indeed, this is true in Newtonian physics), but no changes to the laws of physics would allow for speeds less than zero, not if "speed" was defined in the same way.

 

[peter0302]

Yes, you are right that due to time dilation and length contraction, you can travel seemingly long distances (measured from your pre-launch stationary frame) in a short time, without ever measuring your speed to be greater than C. If we ever get the technology to accelerate a spacecraft to a high fraction of C, we could send spaceships to distant stars and back within the lifespan of the crew. The catch, of course, is everyone who helped build the craft will be long dead when they get back.

The good news is that if the purpose were simply to preserve the human race and seed new planets, it wouldn't matter much. I do, personally, take some comfort in the fact that, if we can just survive a few hundred more years - maybe even less - we'll be able to do so. (Assuming of course we can find suitable planets, which is pretty likely given the discoveries that keep happening).

 

[WillBlake]

Yes, I do agree that the laws of the universe may restrict us to a narrow gap in possible speeds. However, I do not think it is beyond the capability of human science to better understand an underlying cause for this gap. It is quite possible that some intuitive investigation will reveal more on this. Thanks for the educational input.

 

[WillBlake]

There is nothing in the definition of "speed" which limits speeds to c; this limit is not a matter of definition, but rather a consequence of the laws of physics in our universe. It's perfectly possible to imagine a hypothetical universe with different laws where there is no upper limit on speed (indeed, this is true in Newtonian physics), but no changes to the laws of physics would allow for speeds less than zero, not if "speed" was defined in the same way.

The reason I bring up the speed of zero is because it is a "resting speed" simply by definition. If the speed of light was considered the "resting speed", then there would be reason to imagine a speed less than zero. Both of these are limits and can be looked at from either side. I guess my question is, is there math that can elucidate the reason for these limits. Thanks for your input.

 

[DaveC426913]

Mostly I wanted to point out that it is possible to travel to distant star/galaxies within a reasonable time (from the traveler's POV). This doesn't seem to be understood by the mainstream public who tend to think that it would take at least 1 yr to go 1 lightyear etc.

Something that will come in handy for exploring Gliese 581. At a comfortable 1G, the trip takes a mere 6.1 years ship's time while the Earth time is a tolerable 22.6 years. I made note of this in my http://www.davesbrain.ca/science/gliese/gliese_primer.html" .

 

[JesseM]

The reason I bring up the speed of zero is because it is a "resting speed" simply by definition. If the speed of light was considered the "resting speed", then there would be reason to imagine a speed less than zero.

Again, speed cannot be less than zero by the definition of speed itself, so what you're saying doesn't make sense. Speed is always measured in the context of some coordinate system--for example, if we're using a coordinate system where an object is at position x=5 meters at time t=1 second, and then at position x=15 meters at time t=2 seconds, we'd say its speed was 10 meters/second in that coordinate system. If you want to, you can define a coordinate system where a given light wave is at rest, meaning its coordinate position doesn't change over time (though the equations of relativity are only supposed to work in a special class of coordinate systems called 'inertial reference frames', and this wouldn't be one of them so you'd have to write the laws of physics differently in such a coordinate system, and normal rules of relativity like 'nothing moves faster than light' wouldn't apply), but in this coordinate system objects which aren't moving at the speed of light would be moving faster than the light wave, not slower.

 

[peter0302]

Something that will come in handy for exploring Gliese 581. At a comfortable 1G, the trip takes a mere 6.1 years ship's time while the Earth time is a tolerable 22.6 years. I made note of this in my http://www.davesbrain.ca/science/gliese/gliese_primer.html" .

How much fuel would that take?

 

[DaveC426913]

How much fuel would that take?

It would take as many gallons of fuel as there are faeries that can dance on the head of a pin.

 

[peter0302]

I actually was serious. Fission fuel lasts a very long time.

 

[JesseM]

There are some fuel calculations at the bottom of the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html page. It turns out that even if you had a perfectly efficient way of converting fuel mass into forward kinetic energy (even a hypothetical matter/antimatter drive wouldn't do this 100% efficiently), you'd still need 10,000 times more fuel mass than the mass of the payload just to get to the nearest star 4.3 light years away at constant 1g acceleration (and if you wanted to accelerate at 1g for the first half of the trip and then decelerate at 1g for the second half so you could actually stop at your destination rather than zipping by it, you'd need 38,000 times more fuel mass). The basic problem is that if you carry your fuel with you, then at any given point in the trip you have to be burning enough fuel not just to accelerate the payload at 1G, but also to accelerate all the fuel you'll need for the remainder of the trip. So, the most "realistic" proposals for interstellar travel usually sidestep the problem of carrying your fuel with you by having the payload attached to something like a giant sail which is pushed along by a beam sent out from our solar system, either an enormous laser or perhaps something like a beam of tiny pellets (the advantage of the latter is that the pellets could be self-steering to some degree which would cut down on the problem of the beam becoming decollimated and missing the sail). In this case you wouldn't be accelerating at a constant rate though, since the force on the sail as seen in the solar system's frame would be at best constant and more likely decreasing due to decollimation, whereas the force needed to produce a given acceleration on a moving object is greater the larger its fraction of light speed.