Hurried Thoughts: Time Dilation Loop

So, you may have noticed by now that my regular habit is to do very long, intensely researched posts that have been steadily creeping up towards textbook length. It's a bit of imposing, but I usually figure it's the best way to present what are usually very complex and interrelated subject matters. On occasion though, I do have thoughts or make interesting discoveries about simpler subjects that either don't fit in with one of the larger posts I've been working on or would fit only in a post I won't be putting up for a long time. Rather than just sit on these, I figured it might be nice to occasionally throw up a quick post just quickly running over a single idea without worrying too much about polish or context.

 

For this first one, a thought experiment: say you're impatient to get to the future. Maybe you want to see what the technology will be like, maybe you're waiting for a return letter from your pen pal in Fomalhaut, maybe you're just impatient for the next season of Prehistoric Planet, who knows. Say you don't have any way to hibernate or freeze yourself, but you do have some very high-performance spacecraft on hand. So you'll figure you'll move yourself at high velocity, causing time dilation between yourself and home such that you can appear to be moving more quickly forward in time. But you don't want to get far from home, so you can't just fling yourself out into space in a straight line. You could pick some distant point and repeatedly head there and back, but then you'll spend a large portion of the trip at lower velocity as you decelerate at the end of each trip and accelerate at the start of the next one.

 

The best way to maximize your velocity while minimizing your displacement is to move in a circle; reaching your desired velocity and then constantly accelerating towards the center of the circle. This is basically what happens in an orbit, but in this case you're not necessarily orbiting another object, just an arbitrary point in space.

 

But you're going to be feeling that acceleration, and presumably there's only so much you can handle, even if you're trying to minimize the amount of time you're spending on this journey. Thus, if you're aiming for a specific velocity, to achieve a specific rate of time dilation, and you can only handle a certain maximum acceleration, that necessarily implies a particular radius to this circle. How big will that be?

 

In classical mechanics, the radius of the circle is simply linked to an object's centripetal acceleration (towards the center of the circle) and tangential velocity:

 

 

r = circle radius

v = tangential velocity

a = centripetal acceleration

 

But as per usual, special relativity complicates matters. As the tangential velocity approaches the speed of light, the spaceship's momentum increases in such a way that it requires greater acceleration to keep within a given circular path. Going by this guy's numbers (which match up with what I've seen elsewhere), the formula becomes this:

 

c = speed of light

γ = gamma factor

 

Conveniently enough, the gamma factor here is equivalent to the ratio of how quickly time appears to pass outside the spaceship relative to inside; for a gamma of 10, 10 years will pass at home for every 1 year in the spaceship. Thus, if we assume that the center of the spaceship's circle is static relative to home, such that, despite constant acceleration, the spaceship is maintaining constant relative velocity compared to home, we can take this formula to describe the circle's radius from the desired time dilation and maximum acceleration; we don't even have to bother calculating what the tangential velocity ends up being.

 

Thus, for a range of given accelerations and time dilation gamma factors, here are the radii of the resulting circle ("E+" here is short for "*10^" because I copied this table out of Excel):

Orbit Radius (light-years)
	Acceleration (g)
Gamma	0.01	0.1	0.33	1	3	10	100	1000
1.1	20.3361	2.03361	0.61625	0.20336	0.06779	0.02034	0.00203	0.0002
2	290.516	29.0516	8.80352	2.90516	0.96839	0.29052	0.02905	0.00291
5	2324.13	232.413	70.4282	23.2413	7.7471	2.32413	0.23241	0.02324
10	9587.04	958.704	290.516	95.8704	31.9568	9.58704	0.9587	0.09587
100	968291	96829.1	29342.1	9682.91	3227.64	968.291	96.8291	9.68291
1000	9.7E+07	9683864	2934504	968386	322795	96838.6	9683.86	968.386
10000	9.7E+09	9.7E+08	2.9E+08	9.7E+07	3.2E+07	9683874	968387	96838.7
100000	9.7E+11	9.7E+10	2.9E+10	9.7E+09	3.2E+09	9.7E+08	9.7E+07	9683874
1000000	9.7E+13	9.7E+12	2.9E+12	9.7E+11	3.2E+11	9.7E+10	9.7E+09	9.7E+08

 

Not exactly encouraging, is it? For a modest gamma factor of 10 (requiring a velocity of 99.5% c), keeping your circle under a light-year wide requires a continuous acceleration of over 192 g. More reasonable acceleration and more useful time dilation implies quite the trip; past a gamma of a couple hundred you need to start worrying about keeping yourself in the galaxy. You might as well just go visit your friend in Fomalhaut directly.

 

And if you do decide to go on such a trip, you'll take quite a while for each circuit (worked out here by just finding the circumference of each circle and dividing by the velocity as determined by tanh(acosh(γ)) ):

 

Orbit Period, external perspective (years)
	Acceleration (g)
Gamma	0.01	0.1	0.33	1	3	10	100	1000
1.1	306.712	30.6712	9.29432	3.06712	1.02237	0.30671	0.03067	0.00307
2	2107.75	210.775	63.8713	21.0775	7.02584	2.10775	0.21078	0.02108
5	14904.1	1490.41	451.638	149.041	49.6802	14.9041	1.49041	0.14904
10	60540.6	6054.06	1834.56	605.406	201.802	60.5406	6.05406	0.60541
100	6084253	608425	184371	60842.5	20280.8	6084.25	608.425	60.8425
1000	6.1E+08	6.1E+07	1.8E+07	6084554	2028185	608455	60845.5	6084.55
10000	6.1E+10	6.1E+09	1.8E+09	6.1E+08	2E+08	6.1E+07	6084557	608456
100000	6.1E+12	6.1E+11	1.8E+11	6.1E+10	2E+10	6.1E+09	6.1E+08	6.1E+07
1000000	6.1E+14	6.1E+13	1.8E+13	6.1E+12	2E+12	6.1E+11	6.1E+10	6.1E+09

 

But of course it'll be shorter in the spaceship, as that is rather the point here:

Orbit Period, internal perspective (years)
	Acceleration (g)
Gamma	0.01	0.1	0.33	1	3	10	100	1000
1.1	278.829	27.8829	8.44938	2.78829	0.92943	0.27883	0.02788	0.00279
2	1053.88	105.388	31.9356	10.5388	3.51292	1.05388	0.10539	0.01054
5	2980.81	298.081	90.3276	29.8081	9.93604	2.98081	0.29808	0.02981
10	6054.06	605.406	183.456	60.5406	20.1802	6.05406	0.60541	0.06054
100	60842.5	6084.25	1843.71	608.425	202.808	60.8425	6.08425	0.60843
1000	608455	60845.5	18438	6084.55	2028.18	608.455	60.8455	6.08455
10000	6084557	608456	184381	60845.6	20281.9	6084.56	608.456	60.8456
100000	6.1E+07	6084557	1843805	608456	202819	60845.6	6084.56	608.456
1000000	6.1E+08	6.1E+07	1.8E+07	6084557	2028186	608456	60845.6	6084.56

Still, if you want to keep acceleration under 1 g and get back home within your natural lifetime, you're probably not going to be getting more than a few centuries into the future; and that's before accounting for the time just to get up to speed. If you're an immortal who can take a trip of centuries and withstand over 10 g, you might manage millennia.

And, of course, acceleration doesn't come free. Rocket scientists often measure a spacecraft's performance in terms of its achievable change in velocity, or delta-v. Accelerating continuously at 1 g for a full year requires almost 310,000 km/s of delta-v, which is near the limit of what could be achieved with the most powerful speculative designs for antimatter-fueled rockets. Measured in terms of delta-v expended per year passing at home, the trips with the highest gamma and lowest g are most efficient in this respect:

 

Delta-v per External Year (km/s)
	Acceleration (g)
Gamma	0.01	0.1	0.33	1	3	10	100	1000
1.1	2814.29	28142.9	92871.5	281429	844286	2814287	2.8E+07	2.8E+08
2	1547.86	15478.6	51079.3	154786	464357	1547858	1.5E+07	1.5E+08
5	619.143	6191.43	20431.7	61914.3	185743	619143	6191432	6.2E+07
10	309.572	3095.72	10215.9	30957.2	92871.5	309572	3095716	3.1E+07
100	30.9572	309.572	1021.59	3095.72	9287.15	30957.2	309572	3095716
1000	3.09572	30.9572	102.159	309.572	928.715	3095.72	30957.2	309572
10000	0.30957	3.09572	10.2159	30.9572	92.8715	309.572	3095.72	30957.2
100000	0.03096	0.30957	1.02159	3.09572	9.28715	30.9572	309.572	3095.72
1000000	0.0031	0.03096	0.10216	0.30957	0.92871	3.09572	30.9572	309.572

But these are also the longest circuits; as it turns out, the total delta-v required for each circuit with a given gamma factor does not vary with acceleration; but even in the most moderate cases it well exceeds what any vessel bound by the rocket equation could realistically achieve:

 

	Delta-v per
Gamma	circuit (km/s)
1.1	863176.736
2	3262501.4
5	9227747.46
10	18741643.7
100	188351188
1000	1883605121
10000	1.8836E+10
100000	1.8836E+11
1000000	1.8836E+12

And, again, you have to get up to this speed first. A while back I added in some calculations for time dilation under acceleration to the worldbuilding spreadsheet (in the "Unit Converters + Relativity" tab, yes I buried it a bit) for a linear case of travelling between two static locations, and much the same math can be used here:

 

		For 1 g Acceleration			For 10 g Acceleration			For 100 g Acceleration
	Speed	Acceleration time (years)		delta-v	Acceleration time (years)		delta-v	Acceleration time (years)		delta-v
Gamma	(out of c)	(internal)	(external)	(km/s)	(internal)	(external)	(km/s)	(internal)	(external)	(km/s)
1.1	0.416598	0.4295445	0.4437694	132978.4	0.0429545	0.0443769	132978.4	0.0042954	0.0044377	132978.4
2	0.866025	1.2753213	1.6772908	394814	0.1275321	0.1677291	394814	0.0127532	0.0167729	394814
5	0.979796	2.2199548	4.7440947	687253.7	0.2219955	0.4744095	687253.7	0.0221995	0.0474409	687253.7
10	0.994987	2.8985899	9.6353019	897345.6	0.289859	0.9635302	897345.6	0.0289859	0.096353	897345.6
100	0.99995	5.130783	96.833585	1588388	0.5130783	9.6833585	1588388	0.0513078	0.9683359	1588388
1000	1 - 5E-07	7.3605942	968.38379	2278693	0.7360594	96.838379	2278693	0.0736059	9.6838379	2278693
10000	1 - 5E-09	9.5903816	9683.8428	2968991	0.9590382	968.38428	2968991	0.0959038	96.838428	2968991
100000	1 - 5E-11	11.820168	96838.316	3659288	1.1820168	9683.8316	3659288	0.1182017	968.38316	3659288
1000000	1 - 5E-13	14.05002	968448.75	4349606	1.405002	96844.875	4349606	0.1405002	9684.4875	4349606

And, of course, you have to do the same in reverse at the end of the trip to return home.

So, if you want to take any of these trips, you're probably going to have to break a few of the known laws of physics to make it practical, and in general this is by no means an easy way to skip forward into the future. Oh well.

 

A better option may be to find a large black hole and drop into low orbit of that instead; no need to keep accelerating to hold yourself in place while time-dilating, though you still could to get that little extra bit of velocity for more dilation. Either way, you may still have to pay a hefty cost in delta-v and travel time to get into and out of that orbit. But, in keeping with the spirit of this new series, I won't dig into the math for gravitational time dilation today. That's all for now.

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