Energy equivlence for time of travel: Newton and Einstein

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Hobbes_
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Energy equivlence for time of travel: Newton and Einstein

Postby Hobbes_ » Wed Dec 11, 2013 3:55 pm UTC

I've wondered about the limitations relativity (i.e. the light-speed limit) puts on space travel, and felt that there is some offset to the light-speed speed-limit by the fact that space contracts as we put more energy into our travel. And I decided a fair comparison was to compare energy use for a Newtonian and a Relativistic environment. Essentially: if I put some amount of energy into a system in a Newtonian world (I'm assuming instantly, and not worrying about acceleration time) I'll achieve some velocity, and that velocity will get me across some distance in a given amount of time.

Now I put that same energy into a relativistic system. The velocity I achieve isn't as high (and perhaps noticeably different, if the energy/mass ratio is large) so I don't travel that same distance as fast. However, the distance I need to travel (according to my system's frame of reference) is decreased due to the increased velocity. So the time it appears to take to travel that distance (from the moving object reference frame) isn't just the reduced velocity divided by the original distance. I did some math to see what the comparison was, and I was surprised by the result. I was hoping people here could comment on my math and my approach as I am not confident in either. I do static structures for a living: not relativity.

I ran some mathcad code for a Newtonian velocity of 1.5*c, I've tried to translate as best as possible into forum readable code (xn refers to "Newtonian" values and Xr refers to "relativistic" values)

d0 = 1*ly = 9.461e15 m
m0 = 1 kg
c = 2.998e8 m/s

Newtonian:
vn = 1.5*c

tn = d0/vn = 2.104e7 s

En = 1/2 * m0 * vn2 = 1.011e17 J

Relativistic:
Er = En + m0*c2

rho2*c2 = -m02*c4 + (m*c2)2

Er = sqrt(rho2*c2 + (m0*c2)2)

==> MATH <==

vr = c * sqrt(Er2 - c4 * m02) / Er

vr = 2.645e8 m/2 = 0.882 * c

dr = d0 * sqrt(1 - vr2 / c2)

dr = 4.452e15 m = 0.471 * ly

tr = dr / vr = 1.683e7 s

tn / tr = 1.25


So basically, it actually takes less time (as observed by the traveler) to travel in a relativistic universe as compared to a Newtonian one. Thoughts or comments?

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby starslayer » Wed Dec 11, 2013 4:52 pm UTC

You've rediscovered time dilation, basically. For relativity to be consistent, both a boosted frame's time and space coordinates must change if the speed of light is to be the same for all inertial observers. Of course, the traveler does not see time as slowing down; he would say that his internal clock is ticking at the same rate it always did. Instead, he thinks that everyone else is length contracted and that their clocks are ticking slower.

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby davidstarlingm » Wed Dec 11, 2013 5:14 pm UTC

I confess I read your introduction, then skipped your math and went straight to your conclusion. Rude of me, I know. But that was my first thought as well: "Yep, that's time dilation for you."

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby Hobbes_ » Wed Dec 11, 2013 5:27 pm UTC

While I'm perfectly aware of time dilation, I thought it was interesting that the relativistic effects actually make travel faster. The existence of time dilation wasn't enough for me to know that it would take me less experiential time to travel in relativistic universe than a Newtonian one.

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby strake » Wed Dec 11, 2013 5:47 pm UTC

Yep, one can go from A to B in arbitrarily little proper time.

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby Hobbes_ » Wed Dec 11, 2013 5:51 pm UTC

strake wrote:Yep, one can go from A to B in arbitrarily little proper time.


True... but again it doesn't really relate to what I was trying to discover...

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby elasto » Thu Dec 12, 2013 10:40 am UTC

From the point of view of a photon, it arrives at its destination instantaneously. From the point of view of a tachyon - assuming such things exist - it arrives at its destination before it left.

This is why the speed of light is a barrier.

It sounds like you knew of time dilation but didn't fully appreciate what it meant :)

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby Flumble » Thu Dec 12, 2013 2:23 pm UTC

To me it sounds like the people in this thread don't fully appreciate what Hobbes is talking about.

strake wrote:Yep, one can go from A to B in arbitrarily little proper time.

...which goes for both a newtonian and a relativistic universe.


Hobbes is saying that for a certain amount of energy time dilation not only compensates for the speed limit in a relativistic universe, but you'll be faster in it.

I'm curious though: does this apply for every amount of energy?

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ETA: I admit to reading past the maths (it's wibbly-wobbly to me) and simply assuming Hobbes' calculations are right.

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby strake » Thu Dec 12, 2013 3:53 pm UTC

Flumble wrote:To me it sounds like the people in this thread don't fully appreciate what Hobbes is talking about.


Ah, yes, sorry. I learned this when I read it anew.

I'm curious though: does this apply for every amount of energy?


Yes.

In what follows, (√) binds tighterly than (implicit (·)) and (/).

Let (ε, v, t) be (kinetic energy per rest mass, speed, proper time) of our craft, l be proper length of path. Thus

Einstein:
v = √(ε(2+ε))/(1+ε)
t = √(1-v²)l/v = √(1/v²-1)l = l/√(2ε + ε²)

Newton:
v = √(2ε)
t = l/√(2ε)

Indeed, the Newtonian formula is just the Einsteinian formula when ε << 1, as it ought.

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby nitePhyyre » Thu Dec 12, 2013 9:29 pm UTC

strake wrote:In what follows, (√) binds tighterly than (implicit (·)) and (/).

Let (ε, v, t) be (kinetic energy per rest mass, speed, proper time) of our craft, l be proper length of path.
I'm pretty sure it should be tightlyer. :lol:
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Re: Energy equivlence for time of travel: Newton and Einstei

Postby gmalivuk » Fri Dec 13, 2013 1:37 am UTC

That just looks like you're mangling "lightyear".

And standard spelling rules would dictate tightlier, in any case, like friendlier and lovelier.
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Re: Energy equivlence for time of travel: Newton and Einstei

Postby lgw » Fri Dec 13, 2013 8:01 pm UTC

The energy as measured in what reference frame?

For the space traveler the time taken to reach a distant destination is exactly as Newton would predict. The total energy needed to get there as measured in the reference frame of the ship (e.g., what you care about for fuel calculations and such) is exactly as Newton would predict.

Now if your ship doesn't stop at the destination, and zips by at relativistic speed, the twin paradox is at work - each frame (ship and origin/destination) sees the other as having less elapsed time in the journey than their own reference frame, and so the energy used across the trip will be seen differently. If the ship "slows down" and returns to the original reference frame, everyone will agree on the energy used: ship, origin, and Newton.
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Re: Energy equivlence for time of travel: Newton and Einstei

Postby gmalivuk » Fri Dec 13, 2013 8:34 pm UTC

lgw wrote:The total energy needed to get there as measured in the reference frame of the ship (e.g., what you care about for fuel calculations and such) is exactly as Newton would predict.
No, it most definitely isn't. It is possible to get across the galaxy within a single lifetime at no more than 1g of proper acceleration, which is not possible in the Newtonian case.
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Re: Energy equivlence for time of travel: Newton and Einstei

Postby lgw » Fri Dec 13, 2013 10:14 pm UTC

gmalivuk wrote:
lgw wrote:The total energy needed to get there as measured in the reference frame of the ship (e.g., what you care about for fuel calculations and such) is exactly as Newton would predict.
No, it most definitely isn't. It is possible to get across the galaxy within a single lifetime at no more than 1g of proper acceleration, which is not possible in the Newtonian case.


Well, I've discovered the math is now beyond me, so I'll take you at your word (well, at your assumed math). It seems bizarre that you can get basically anywhere in 100 years at 1 g (or rather, bizarre that I don't remember that from school, but then, birds weren't dinosaurs back then).
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Re: Energy equivlence for time of travel: Newton and Einstei

Postby gmalivuk » Fri Dec 13, 2013 10:49 pm UTC

Here are the relevant equations.

As I recall from making the same mistake myself, the Newtonian results come from doing something wonky with the acceleration, like proper meters per proper second per stationary-observer second. My workthrough and realization of the mistake are somewhere on the forums from 2007, but the search seems to be going super slow at the moment.

Edit: Here's the post where I realized my mistake. I was apparently using stationary-observer meters per proper seconds^2. Which corresponds to an ever decreasing proper acceleration.

It's interesting, looking at my posts containing the word "acceleration", that I had almost fixed that particular misconception four months earlie, when disagreeing with Yakk's independently-arrived-at belief that the effects all balanced out and gave you Newton in the end, but then I re-convinced myself that this was indeed the case.

I suspect that it's a conclusion a lot of people come to on their way toward understanding SR, because even now I couldn't tell you where the hyperbolic trig functions come from in the correct equations but I had no problem re-running the calculations I used to arrive at the wrong ones.
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Re: Energy equivlence for time of travel: Newton and Einstei

Postby strake » Mon Dec 16, 2013 2:50 pm UTC

gmalivuk wrote:That just looks like you're mangling "lightyear".

And standard spelling rules would dictate tightlier, in any case, like friendlier and lovelier.


But "friendly" and "lovely" are adjectives, and "tightly" is an adverb.

I wondered myself how to form a comparative adverb, and ultimately just unilaterally determined that "betterly" sounds better than "weller" as such ☺

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby Twistar » Mon Dec 16, 2013 6:43 pm UTC

gmalivuk wrote:Here are the relevant equations.

As I recall from making the same mistake myself, the Newtonian results come from doing something wonky with the acceleration, like proper meters per proper second per stationary-observer second. My workthrough and realization of the mistake are somewhere on the forums from 2007, but the search seems to be going super slow at the moment.

Edit: Here's the post where I realized my mistake. I was apparently using stationary-observer meters per proper seconds^2. Which corresponds to an ever decreasing proper acceleration.

It's interesting, looking at my posts containing the word "acceleration", that I had almost fixed that particular misconception four months earlie, when disagreeing with Yakk's independently-arrived-at belief that the effects all balanced out and gave you Newton in the end, but then I re-convinced myself that this was indeed the case.

I suspect that it's a conclusion a lot of people come to on their way toward understanding SR, because even now I couldn't tell you where the hyperbolic trig functions come from in the correct equations but I had no problem re-running the calculations I used to arrive at the wrong ones.


To get the trig functions you have to write down the four-velocity and four acceleration in some "stationary" frame and in the instantaneous rest frame of the accelerating object. Then using a couple properties about the four acceleration that you know are true (including the fact that it's a four vector so it's minkowski length is conserved) you get differential equations relating the space and time components of the four position which are solved by cosh and sinh. The Cosh and Sinh make sense because the slope of the world line must approach but not exceed the slope of the light line. What's confusing is that constant four acceleration doesn't mean the particles velocity increases at a constant rate. Of course it is impossible for a particles velocity to always increase at a constant rate because the particle would exceed the speed of light at some point.

One interesting result here is that if you start far enough away from a light source and undergo this type of constant four acceleration away from the source it is possible that if the starting distance is large enough and you are accelerating fast enough the light will never catch up to you, even though it is always travelling faster than you.

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Re: Energy equivlence for time of travel: Newton and Einstei

Postby lgw » Tue Dec 17, 2013 9:16 pm UTC

Twistar wrote:...

One interesting result here is that if you start far enough away from a light source and undergo this type of constant four acceleration away from the source it is possible that if the starting distance is large enough and you are accelerating fast enough the light will never catch up to you, even though it is always travelling faster than you.


I thought that was only true of very high acceleration (effectively, acceleration and gravity are the same for black holes too). Can that happen at 1 g for a distant light source? At the acceleration of the Earth in it's orbit?

I was recently surprised to realize just how much "time parallax" there is from Earth orbital motion: as the edge of the visible universe, the point in the age of the universe that we see the light from changes by hundreds of millions of years just from the minor shift in velocity across the seasons. Would there be a similar "light can't catch you" effect for light originating billions of light years away with low acceleration, or is that strictly a very high acceleration thing?
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Re: Energy equivlence for time of travel: Newton and Einstei

Postby gmalivuk » Tue Dec 17, 2013 9:46 pm UTC

There is nothing special about different amounts of acceleration.

Any constant acceleration creates an event horizon opposite the direction you're accelerating, beyond which no light emitted after you start accelerating will ever reach you while you continue accelerating. That distance is about 1 light year if you're accelerating at 1g. It's 100 light years if you're accelerating at 0.01g, but it's still there all the same.
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Re: Energy equivlence for time of travel: Newton and Einstei

Postby doogly » Tue Dec 17, 2013 11:27 pm UTC

It is called a Rindler Horizon
http://en.wikipedia.org/wiki/Rindler_co ... er_horizon

It even radiates!
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Re: Energy equivlence for time of travel: Newton and Einstei

Postby lgw » Wed Dec 18, 2013 10:47 pm UTC

gmalivuk wrote:There is nothing special about different amounts of acceleration.

Any constant acceleration creates an event horizon opposite the direction you're accelerating, beyond which no light emitted after you start accelerating will ever reach you while you continue accelerating. That distance is about 1 light year if you're accelerating at 1g. It's 100 light years if you're accelerating at 0.01g, but it's still there all the same.


Ahh, yup, that makes perfect sense when I consider that a black hole with 1 g of Newtonian-gravity at the event horizon has a "radius" of about 1 light year (and so on with the other numbers). But of course we don't see this effect from the Earth's orbital acceleration because the direction isn't steady.
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