-- By the Physicist blog Physics Relativity

Q: In relativity, length contracts at high speeds. But what’s contracting? Is it distance or space or is there even a difference?

The original question was: I can’t discover a consistent answer to this query; please help.  A spaceship leaves Earth and heads to a star four mild years away at 80% of light velocity.  An observer on Earth knows that the spaceship’s clock will run slower than his clock by 40% for the whole thing of the journey (in response to the Lorentz formulation).

In line with the Earth-based observer, the spaceship will arrive at the star in 5 years.  Nevertheless, due to time dilation, the spaceship’s clock will only learn 3 years of elapsed time on arrival.  To an astronaut on the spaceship, the distance to the star appears to be simply 2.four mild years as a result of it took him simply 3 years to get there whereas touring at 80% mild velocity.

This example is typically explained as a consequence of length contraction.  But what is it that’s contracting?  Some authors put it right down to space itself contracting, or just distance contracting (which it seems to me amounts to the identical thing) and others say that’s nonsense because you might posit two spaceships heading in the identical path momentarily aspect by aspect and traveling at totally different speeds, so how can there be two totally different distances?

So what is the right solution to understand the state of affairs from the astronaut’s perspective?


Physicist: Space and time don’t react to how you move around.  They don’t contract or decelerate just because you move quick relative to somebody someplace.  What modifications is the way you perceive space and time.

There’s no true “forward” course and, terrifyingly, there’s no true “future” path or even “space but not time” path.  All of these instructions and the lengths of issues in these instructions are subjective and even, dare I say, relative.

Whenever you measure the length of something in space (in different words, “normally”), the full length isn’t just the length in the x or y directions, it’s a specific mixture of both that works out precisely the best way you’d assume it should.  Once you measure the length of something in spacetime, the whole length isn’t simply the length in the space or time instructions, it’s a specific mixture of each that works out in additional or much less the other of how you’d assume it ought to.

We don’t speak concerning the three dimensions of space individually, because they’re not likely distinct.  The ahead, right, and up directions are a good approach to describe the three totally different dimensions of space, but in fact they range from perspective to perspective.  Simply name someone from the other aspect of the planet and ask them “What’s up?” and you’ll end up immediately embroiled in irreconcilable battle.  Everyone can agree that it’s straightforward to select three mutually perpendicular instructions in our three-dimensional universe (attempt it), but there’s no sense in making an attempt to specify which specific three are the “true” instructions.

When you insist on measuring issues in just one course, then totally different views will end in totally different lengths.  To seek out the whole length, d, requires doing a couple measurements, x and y (and z too, in 3 dimensions), and making use of some Pythagoras, d2=x2+y2.

A meter stick is a meter lengthy (therefore the identify), so in case you place it flat on a table and measure its horizontal length (with a… tape measure or something), you’ll discover that its horizontal length is 100cm and its vertical length is zero.  Provided that, you possibly can fairly divine that it have to be 100cm long.  But should you tilt it up (or equivalently, tilt your head a bit), then the horizontal and vertical lengths change.  There’s nothing profound occurring.  To handle a universe merciless sufficient to permit such differing views we use the “Euclidean metric”, d^2=x^2+y^2+z^2, to seek out the entire length of issues given their lengths in every of the varied directions.  The length in any given path (x, y, z) can change, but the complete length (d) stays the identical.

Einstein’s massive contribution (or certainly one of them at least) was “combining” time and space underneath the umbrella of “spacetime”, so named as a result of Germans love sticking words together in a conventional course of referred to as (roughly translated) stickingwordstogethertomakeonereallylongdifficulttoreadandoftunpronounceableword.

The totally different spatial dimensions are equal.  To see for yourself, stroll north and south, then walk east and west.  Until you’re carrying a compass, you shouldn’t notice any distinction.  But clearly time is totally different.  To see for your self, first walk north and south, then walk to tomorrow and back to yesterday.  So when someone cleverly volunteers “we live in a 4 dimensional universe!”, they’re being a little imprecise.  Physicists, who love precision slightly greater than being understood, choose to say “we live in a 3+1 dimensional universe!” to clarify that there are three space dimensions and one time dimension.

But while time and space are totally different, they’re not utterly separate.  In very a lot the same means that the forward course varies between views, the “future direction” additionally varies.  And in the identical means that rotating views exchanges instructions, shifting at totally different velocities exchanges the time path and course of motion.  The full “distance” between factors in spacetime is referred to as the “interval”, L.  For people conversant in the Euclidean metric, the “Minkowski metric” should look eerily familiar: L^2=x^2+y^2+z^2-(ct)^2=d^2-(ct)^2.  Some people will flip the sign on this, L^2=-x^2-y^2-z^2+(ct)^2=-d^2+(ct)^2, as a result of it makes it a little easier to speak concerning the time skilled on a specific path (the truth is, I’m gonna do this in a minute), but the necessary thing is not the sign of this equation, it’s that it’s fixed between totally different views.  It ought to hassle you that L^2 might be damaging, but… don’t fear about it.  It’s positive.

When you’re wondering, the spacetime interval is a direct consequence of rule #1 in relativity: the velocity of light is the identical to everyone.  The brief approach to see this is to note that in the event you find the interval between the beginning and end factors of a mild beam’s journey, the interval is all the time zero because fracdt=cRightarrow d=ctRightarrow d^2=(ct)^2Rightarrow d^2-(ct)^2=0.  The lengthy solution to see why the interval is what it is, is a little long.

There are two issues to note concerning the spacetime interval.  First, that “c” is the velocity of sunshine and it principally offers a unit conversion between meters and seconds (or furlongs and fortnights, or whatever models you favor for distance and time).  So 1 second has an interval of about 300,000 km (one “light-second“), which is a lot of the distance between right here and the Moon.  It seems that the velocity at which mild travels comes from the “c” in this equation.  So the velocity of sunshine is dictated by the nature of space and time (as described by the Minkowski metric), not the other approach around.  Which is good to know.

Second and more essential is that unfavourable.  That basically screws things up.  It is arguably liable for damn-near all the bizarre, unintuitive stuff that falls out of special relativity: time dilations, length contractions, twin paradoxes, Einstein’s haircut and marriages, every thing.  In specific (and this is why the trade between distance and period is so unintuitive), if d^2-(ct)^2 is fixed, then when d will increase, so does t.

This is in stark distinction to common distance, where if x^2+y^2 is fixed, a rise in x means a decrease in y.  Picture that in your head and it is sensible.  Image relativity in your head and it doesn’t.

Left: The points a distance of 1 away type the origin type a circle.  The 2 blue strains are the same length.  Right: The points a spacetime interval of 1 away from the origin type a hyperbola.  The 2 purple strains are the identical “length”.  Here time is the vertical axis and one of many space directions is the horizontal axis.  So for those who sit still you’d hint out a path like the primary purple line and in case you have been shifting to the correct you’d trace out a path just like the second purple line.

Now brace yourself, as a result of right here comes the point.  The original query was about a journey that, from the attitude of Earth, was d = 4 light-years long, at a velocity of v = 0.8c, and taking t = 5 years.  The great thing about utilizing “light units” (light-years, light-seconds, and so forth.) is that the spacetime interval is very easy to work with.  The interval between the launch and landing of the spaceship is:

L^2=-d^2+(ct)^2=-(4)^2+(5)^2=-16+25=9

So the interval is L = 3 light-years.

Left: Earth and an alien world sit still (travel by way of time but not space) four light-years apart whereas a spaceship traveling to the best at v=0.8c takes 5 years to travel between them.  Right: A spaceship sits nonetheless (travels by means of time however not space) for 3 years while Earth and an alien world journey to the left at v=zero.8c.

Like common distance, the facility of the spacetime interval is that it is the identical from all perspectives.  From the attitude of the spaceship the launch and touchdown occur in the same place.  It’s like a narcissist on a practice: they get on and get off in the identical place, whereas the world moves around them.  So d = zero and it’s just a query of how much time passes:

3^2=-0^2+(ct)^2Rightarrow 3^2=(ct)^2Rightarrow 3=ct

So, t = 3 years as a result of 3 light-years divided by the velocity of light is 3 years.

So identical to altering your perspective by tilting your head modifications the horizontal and vertical lengths of stuff (whereas leaving the full length the identical), changing your perspective by shifting at a totally different velocity modifications length-in-the-direction-of-motion and period (while protecting the spacetime interval the same).

That’s time dilation (5 years for Earth, but three years for the spaceship).  Length contraction is a little extra delicate.  Usually once you measure something you get out your meter stick (or yard stick, depending on where you reside), put it next to the thing in question and growth: measured.  But how do you measure the length of stuff if you’re shifting previous it?  With a stopwatch.

How are you going to tell that mile markers are a mile aside?  Because once you’re driving at 60 mph you see one a minute.

So, like the unique query pointed out, if it takes you three years to get to your vacation spot, which is approaching you at zero.8c, then it have to be three×0.eight = 2.4 light-years away.  Discover that in the diagram with the planets above, on the left they’re 5 light-years aside and on the correct they’re 2.4ish light-years aside (measure horizontally within the space path).

It looks like length contraction must be extra difficult than this, however it’s actually not.  You will get yourself tied in knots enthusiastic about this too exhausting.  In any case, if you speak about “the distance to that whatever-it-is” you’re talking about a straight line in spacetime between “here right now” and “over there right now”, however “now” is a little slippery when the “future direction” is relative.  Fortunately, “time multiplied by speed is distance” works effective.

There are a few methods to look at the state of affairs, however all of them boil right down to the same huge concept: views shifting relative to each other see all types of issues totally different.  Actuality itself, space and time and the stuff in it doesn’t change, but how we view it and interact with it doesn’t fairly comply with the principles we imagine.