# What does special relativity say about the apparent versus real length contraction in regards to a large body such as Earth?

Suppose there are 2 spaceships moving perpendicular to a large body (ex. Earth) at very high speeds. According to all official educational resources I have found online ( such as Fermi Lab and many more ), the length contraction and time dilation effects of special relativity are said to be entirely real (ex. A spaceship moving near c actually has contracted in length and twins actually experience time dilation in the twin paradox).

This would mean if a spaceship travels perpendicular to another spaceship relative to Earth, Earth would have to take on a pancake shape in relationship to each spaceship (the view from each spaceship that Earth has become a pancake like object is claimed to be a real phenomenon - just as the spaceships would really experience length contraction as viewed from earth). Hence, Earth would have to be both a horizontally and vertically oriented pancake at the same time as both spaceships are moving near the speed of light at the same time.

How could this appearance be said to be real - it insists the Earth is in multiple pancake shapes simultaneously? If Special Relativity simply states what things appear as to the observer then this result isn't obviously contradictory. But all the major educational outlets I see online insist these results are entirely real (for example FermiLabs insistence that the twin paradox is real - one twin really is younger).

I have used length contraction in this example because it is more "visual" than time dilation. What seems to be impossible contradictions of special relativity when its effects are viewed as real (not just apparent) seems easier to think about.

When two rockets pass Earth in different directions, they will each observe distances to be contracted in their respective directions of motion, but there is no contradiction. In neither case is the Earth itself affected. It is simply described differently in different reference frames.

To further complicate the situation though, observations in Special Relativity are not what is seen directly, but rather what is deduced after correcting for the finite travel-time of light from the observed object. Ironically, this effect would result in the Earth appearing round (but slightly rotated) to observers in both rockets. Look up the Penrose-Terrell Effect for more details.

• So the effects of Special Relativity are Apparent as opposed to Real? Is that what this answer states? Thank you for your help, David Commented Oct 7, 2019 at 21:26
• @prblmSlvr You can't say the results aren't real. Each observer's description has an equal standing, even though they are different. Commented Oct 7, 2019 at 21:34
• To try to clear up the communication issue here, when a physicsit says that the effects are "real" they don't mean "length contraction is the same as a giant vice crushing the contracted thing", and do mean, "the observers correctly measure different lengths because they are using different definitions of simultaneity". Commented Oct 7, 2019 at 21:46
• So why is the twin paradox real in the sense that one twin really has aged slower than the other when they meet back on earth? This seems to be stating that time dilation is not an apparent effect but an entirely real effect? Commented Oct 7, 2019 at 22:05
• @garyp This reference explains why length contraction wouldn't be directly visible in a photograph (for example): math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html Commented Oct 8, 2019 at 0:59

Length contraction is real in the sense that if you use the same rulers and clocks you use to measure everything else in your moving spaceship, then those same rulers and clocks will tell you that the Earth really is contracted. However, Special Relativity says that not everyone uses the same clocks and rulers, and this is why citizens on Earth can equivalently claim and measure that the Earth is not contracted.

Let’s forget relativity and motion think in terms of just space. Say you and your friend are next-door neighbors, and want to construct a tunnel between your houses. You start your floor plan by constructing a coordinate system, laying down a grid a rulers at right angles representing the x and y coordinates. You might prefer your grid to lie at right angles with respect to your furniture, walls, so you can record their locations easily with coordinates like (0, 0) and (5, 0) as opposed to (0, 0) and (3, 4). Suppose you divided up the work and told your friend to do the same for his house, and then the two of you can stitch your floor plans together to determine where the tunnel should be built. However, your friend’s house is at an angle to yours, and when he comes over his coordinates are all rotated with respect to yours! Worse, he seems to only pay attention to one coordinate at a time. “My room is five meters long!”, he says. “How are you measuring it to be four meters long one way, and three meters the other way? That isn’t five, my room is longer than that!” Of course, you know geometry, and aren’t surprised nor are you worried that his room is being compressed.

The same thing happens in the spaceship scenario. We understand that standing at relative orientations to each other means we see objects from different perspectives - the dimensions of space rotate into each other when we twirl around. Special relativity says the same thing applies to relative velocities - the dimensions of space and time rotate into each other. If we limit ourselves to thinking in terms of lengths without accounting for times, then we might end up like our confused friend. Special relativity teaches us that the entire picture exists not in space, but spacetime. However, because we experience time differently from space, spacetime rotations are as unintuitive to us as ordinary rotations are to our friend. With practice however, we can learn to translate our observer-dependent measurements of space and time into observer-independent descriptions of objects in spacetime.

The effect is, in fact, "real" -- but measuring it fundamentally requires a sense of simultaneity.

To see this, think about how you measure "length" when you're not right on top of the object in the first place. Let's say you run past a long building; how might you measure how long it is? Well, if you know your speed, you could just time how long it took to run from one side to the other... except, oops, that method brings time dilation into play. (And so must any other similar method).

It's easier to see with a smaller object. Let's take the classic example of the garage-door paradox.

Imagine we have Alice and Bob, taking a break from testing encryption protocols to take up relativistic pole vaulting. Alice, carrying a 10 m pole, runs at a garage with doors on either end, that is only 6 m long. Somehow or other, she's attained a speed of $$\,^4/_5$$ of $$c$$ -- so from Bob's perspective, she's been length-contracted down to $$\Delta x \sqrt{1-(v/c)^2}= 10\,m \sqrt{1-0.8^2} = (10\,m) 0.6 = 6\,m$$, and nicely fits in the garage.

Bob, therefore, stands with the front door open and the back door closed. When Alice enters the garage, there is a single moment where she is entirely contained inside the garage. At the instant, Bob closes the front door and opens the back door, letting her through. At no point are the two doors open at the same time.

I think, in this case, it's very easy to see that the length contraction effect is "real" -- Alice has just managed to fit her 10 m pole in a space almost half as big! However, it sure seems like this scenario should create a paradox once we examine things from Alice's point of view.

After all, from Alice's point of view, she's standing still, and Bob is the one coming at her. The garage, too, is length contracted -- now she's trying to fit her 10 m pole in a garage only 3.6m long! Surely she must crash through one or the other door?

Well, no, because her lines of simultaneity aren't the same as Bob's. Two events that are simultaneous, but separated by a distance of 6m, in Bob's frame, are separated by both time and space in Alice's frame -- the distance has diminished to 3.6m, but the time between the two events has gone from $$0$$ to $$\Delta x \frac{(v/c)}{\sqrt{1-(v/c)^2}}=(6 m) \frac{0.8}{0.6}=2.94/c$$. This is a fantastically small amount of time, but Alice is going fantastically fast, so this is more than enough time for her to slip through. Note, by the way, that each individual event is in fact identical across frames -- Alice will still perceive the back door opening at the precise moment her pole touches it, and still perceives that front door closing at the precise moment her pole crosses it; it's just that those events have now been separated in time as well as space.

The proper way to think about such things is to imagine it as a rotation of sorts, in hyperbolic spacetime:

In the inertial frame of the object itself (such as the pole or the garage), an observer measures quantities as represented in the graph on the left. Vertical axis is time, horizontal axis is space, the blue region represents the spacetime "volume" the object -- let's say it's the rod -- occupies. You can see that it's "square" vertically -- i.e. it isn't moving. Measuring the length of the rod amounts to measuring the distance between the two points $$(0s,0m)$$ and $$(0s, 10m)$$ -- where the first entry is the time of the event and the second the spatial location. This is represented by the red line on the left.

However, from Bob's perspective -- the graph on the right -- this is an incorrect measurement. You see that the the blue region, representing the pole, is tilted and squashed; in the process of making both ends of the pole move (shift rightward as time passes), we've had to push the ends together to ensure that the speed of light remains constant. So from Bob's perspective, the two events that Alice is using to measure the length of the pole -- the red line on his grah -- are separated not only in time but also in space; so from Bob's perspective, Alice is measuring the distance between "where the back end of her pole was at $$t=0$$" and "where the front end of her pole will be at a later point in time", and is obviously getting a much larger number than she should! If Bob measures the length of the pole in his reference frame, he will again pick two points that are separated purely in space -- but he'll be measuring the much smaller length in blue, and get a number much smaller than Alice will. (Symmetrically, Alice also thinks that Bob is measuring an incorrect length; she sees him as comparing the points on her blue line on the left. Though it appears longer than the red one, that's because I can't help but graph this in Euclidean space; spacetime is in fact non-Euclidean and if you measure the length of the line according to the metric you will get (the same) smaller number.)

Again, this length is real. You can fit an object into a container that shouldn't fit it, in the same way that a particle that should decay in microseconds can last for almost an entire second close enough to the speed of light. But at the same time, it is a matter of perspective. Time, space, duration, and length are all relative; only the spacetime equivalents, the ratios and angles between, are invariant.

• So this is stating that the length contraction is absolutely real? The a ladder of length 10m can fit inside a garage of length 6m because it really has contracted. Doesn't this contradict with the statement that D Halsey shared that physicists don't mean a vice crushing the contracted thing? Commented Oct 8, 2019 at 3:04
• Yes to the first, and no to the second. If you measure the actual forces on the ladder (with the right relativistic formalism), you're not going to see anything that looks like a crushing force -- the ladder is not feeling any strain it wouldn't normally. Space itself has contracted 'underneath' it, but in the ladder's own reference frame nothing has changed -- it's in the same inertial frame it always is. Commented Oct 8, 2019 at 5:25

## What does "real" mean?

Real means that differences in the measurements made by observers in relative motion are not attributable to an error on the part of either (or both) observers.

This applies to both length contraction and time dilation.

## Both time-dilation and length-contraction are symmetric.

When we are in relative motion, I will see you compressed along the direction of motion and see your watch running too slow.

And you will make the same observations about me.

## But wait . . . the twin paradox leads to a persistent difference!

The first thing to understand about this is that it is a different issue from plain ole time dilation. Time dilation means I see observer time passing too slowly for you and vice versa. Neither one of us is more right than the other. The conflict between these two points of view is a non-issue for purely inertial observers because they are only ever in the same place at the same time once, so that can't make a local comparison of how fast their clocks are ticking.

But if we come to relative rest, then we each observe the other's clock running normally again. There is nothing permanent about plain ole time dilation.

### So why is the twin paradox different?

In the paradox one twin follows an inertial path, and the other does not, so the situation is not fully symmetric and you don't expect a symmetric outcome.

### A digression

Let's start by thinking about plane geometry. Consider a triangle. We're at vertex A and we want to get to vertex B. There are two ways we can go: the direct path and the path through vertex C. The direct path is always shorter.

In analytic geometry you can blame that on the nature of distance calculation $$(\Delta s)^2 = (\Delta x)^2 + (\Delta y)^2 \;.$$ If you line your triangle up with your coordinate system so that the straight path is along $$y$$, and mirror the second segment of the indirect path, then you can say

"Ah ha! That path has the same $$\Delta y$$, but also a contribution from $$\Delta x$$ so it must be longer!"

$$(\Delta s)_{AB} < (\Delta s)_{AC} + (\Delta s)_{CB} \;. \tag{Triangle inequality}$$

### What does that have to do with the twin paradox?

It turns out that the time experiences by an observer between two events is a kind of geometric distance measurement, just like the distance you have to travel on the triangle, and just like with the triangle different paths can have different totals.

The wrinkle is that (only moving in one spatial dimension), the distance rule in Einstein's universe (i.e. the one we really live in) is $$(\Delta s)^2 = (c \,\Delta t)^2 {\color{red} -} (\Delta x)^2 \;.$$

The minus sign is crucial. We'll come back to it.

This distance rule with the minus sign is the mark of "Minkowski space-time". (The minus sign is also a bone of notational contention. Some people would switch the order of the two symbols. This order is most convenient for the discussion at hand.)

### Lorentz invariants

Now here's a surprise: all inertial observers will agree on the value of $$(\Delta s)^2$$ along a path. This value, called "the interval" is something that everyone can agree upon.

The reason for this is that each observer sees a different time interval and length interval, but any increase in one is always canceled out by a increase in the other. You can prove this to yourself for a straight line motion by plugging in the Lorentz transform long hand. It is a little tedious, but worth it.

### Analytic geometry and Minkowski space-time.

Any inertial observer can say

"In my coordinate system I spent $$\Delta \tau$$ proper-time and didn't go anywhere in space, so $$(\Delta s)^2 = (c \, \Delta \tau)^2$$ or $$\Delta \tau = \frac{\Delta s}{c} \;.$$

And everybody can agree on that.

But what about an observer who was there at the start and end of our inertial observer's trip, but popped out to the shops in the meantime? Using the same argument we used for the triangle inequality we can conclude that the travelling twin spent less time on the journey.

## So, what is the equivalent for length contraction?

There isn't one. Or at least, not without going faster than light.

You notice that we defined the interval in terms of a square? And that we compute it with a difference?

The interval can have either sign, and when it has a negative sign $$\sqrt{-(\Delta s)^2}$$ is the proper length between two points in space-time.

So in principle, you can say

"Oh, this route between those two events is shorter than that one. Ta da!"

except that the routes have to have (at least one leg for which) $$\Delta x > c \, \Delta t$$ which is to say faster than light motion.

• @PM2Ring Typos have been addressed. Commented Jul 4, 2023 at 18:50

I have used length contraction in this example because it is more "visual" than time dilation."

I think that's your problem. I don't think you can un-link the spatial transformations from the temporal.

From your point of view, on board Space Ship 1, if you want to measure the shape of the Earth, you must choose several different points on Earth and you must somehow (e.g., by using radar) figure out where each of those points was in your coordinate system at one particular instant in time. If you don't get them all at the same instant, then your measurement will be all wrong because, from your point of view, It's the Earth that's moving extremely fast.

The problem is, I am on board Space Ship 2. I am moving in a different direction, and I have made my own measurements. If we compare notes with each other, You and I will be unable to agree upon what "the same time" means. The events (measurements) that you said were simultaneous appear to me to have happened at different times; and the events that I say were simultaneous will seem to have happened at different times in your coordinate system.

• Suppose the spaceships are moving "forever" perpendicular to Earth near the speed of light. I don't understand why agreeing on time must come into this thought experiment. Regardless of when you ask the spacetravelers, they would say the Earth looks like a pancake ( solely based on special relativity's effects ). Now do trained physicists state this view is Real or Apparent? Thank you for your answer - I need some more clarification about why time dilation must also be considered in this example as I don't see why it is necessary but surely would like to understand if that is in fact the Commented Oct 7, 2019 at 21:39
• @prblmSlvr, You would say that the Earth looks like a pancake, flattened one way. I would say that the Earth looks like a pancake, flattened a different way. Neither one of us would have any way of knowing its "true" shape (i.e., its shape in its own coordinate system) if we did not know the Special Theory of Relativity. That much is "real." Commented Oct 7, 2019 at 21:47
• @prblmSlvr, What does it mean to "know the shape" of a thing? It means knowing the spatial relationships between different points on the thing. That's easy to do if you locate the points in the thing's own coordinate system. But if you are flashing past in a space ship, Your only option is to locate the points in your own coordinate system. But, they're moving in your coordinate system, so you have to locate all of them in the same instant. The reason why you and I can't agree on the Earth's shape when we are in different ships, is that we can't agree on what "the same instant" means. Commented Oct 7, 2019 at 21:50
• @prblmSlvr, P.S., I bet that if you really understood the ladder paradox (a.k.a., the "barn and pole paradox"), then this shape-of-the-earth paradox would no longer be a mystery for you. Commented Oct 7, 2019 at 21:53

Suppose you had a train passing a station at 20 metres per second, and all along the platform were numbered marks at metre intervals like a giant ruler. One way you could measure the length of the speeding train is to have video cameras at each end of the platform and compare the images shown on them at a given time t. If the camera at one end of the platform showed the nose of the train to be next to mark 683 metres at time t, while the camera at the other end showed the tail of the train to be alongside mark 71 metres at time t, you could calculate that the length of the train was 612 metres, simply by subtracting the two readings.

However, let's suppose your camera clocks were out of synch, with the camera at the rear running a second slow. In that case, you would be comparing the positions of the front and rear of the train at different times. Specifically, you would be comparing the position of the front of the train with its position at the rear a second later, during which time the train will have moved forward 20 metres, so your calculation would under-estimate the length of the train by 20 metres, or, to put it another way, the train would seem length contracted by 20 metres.

That is actually how length contraction comes about. An object in its rest frame exists in the moment, so in the case of a train the front and rear of the train exist at the same moment in time. However, in a frame in which the train is travelling, the 'now' at which the two ends of the train exist in its rest frame are two quite separate moments of time in the moving frame, and that is why the train is shorter in the moving frame, because the rear of the train has moved forward towards the front in the time interval.

A very simple example can be given illustrating how two observers can view the very same physical objects at the same time and see apparent conflicting dimensions and/or shapes. No motion or relativity effects are necessarily required.
Let two observers each have a foot long ruler so that when they hold up their rulers aligned and overlaid with each other, each observer concludes that the opposing ruler is the same length as their own. Now let the observers move 10 feet apart and repeat the measurements. Each holds up their ruler in common visual alignment with the other so that each one can again view their own ruler aligned and overlaid with the other observers remote ruler. Each now observes the opposite ruler to be shorter in length than their own ruler. The separation distance has shrunk the "apparent dimensions" of the remote rulers for each observer. The observers are now looking at identical objects and reaching entirely different conclusions as to their relative sizes, i.e. which one is longer.
In a similar fashion, observers on space ships passing in opposite directions at high speed could be holding up foot rulers for comparison as they pass and see entirely opposite results as to which is longer. In that case the reason would be the relativity induced contractions due to speed differences but the observed effects are real in both examples. If the space ships stop or travel together the rulers would regain equal length appearances to the observers on each ship, just like they would if the first example observers remove the separation distance.