# Distance between footsteps

## Will his footprints be abnormally far apart from each other?

4 vote(s)
57.1%

2 vote(s)
28.6%
3. ### I don't know

1 vote(s)
14.3%
1. Sep 29, 2014

### Jaams

A man suddenly decides to walk nearly at the speed of light. Without changing the length of his steps, will he need fewer steps to reach his destination?

2. Sep 29, 2014

### A.T.

Length of his steps in which frame of reference? His own, or that of the surface?

3. Sep 29, 2014

### Jaams

From his own perspective. All the man does is increase his pace.

4. Sep 29, 2014

### pervect

Staff Emeritus
Rigid bodies don't exist in relativity, but there is something called "Born rigidity" that does. Unfortunately, Born rigidity can't be defined for rotating objects, and the normal walking motion requires the thigh bones to rotate back and forth during the walk. Therefore I don't think there is any simple idealized sort of "walking motion" that one can define based on Born rigidity.

I believe your notion of "just increases his pace" implies the idea that the mans bones are rigid, but the only available relativistic definition of "rigid" doesn't apply here. So I don't see how it is possible to give even a theoretical answer to your question unless you can define some notion of how a man with non-rigid bones "walks".

[add]For instance part of the notion of a constant stride would be based on the length of the leg. But having a constant length leg requires the leg to be rigid.

5. Sep 29, 2014

### pervect

Staff Emeritus
I'll propose a different question that I can answer that may illuminate some of the other relativistic issues besides rigidity.

A man rides a bicycle at close to the speed of light. There is a tack on the tire that makes marks in the ground every revolution. As the man rides faster, the wheel expands, but he compensates for this by having an assistant measuring the length between tack marks in the cyclist frame after he gets up to speed.

(This is how we get around the rigidity issue - we don't attempt to say that the wheel is rigid, which is impossible. Instead we assume it's not, and just measure how much it stretches to get the rigidity issue out of the problem.)

(add: You can do the same trick with your walker, just have his assistant measure the stride length. I find it easier to imagine a bicyclist moving near the speed of light than a walker, but if you're dead set on having a walker, I suppose you can do it, as long as you measure his stride)

If the proper distance between the start and finish lines is L, and the proper distance between tack marks in the bicyclist frame is l, how many tack marks are on the ground when the bicyclist finishes?

The answer will be L / (l gamma), which is easiest to see in the cyclist frame. Here gamma = 1 / sqrt( 1 - v^2 / c^2). The distance to the destination will be L / gamma in that cyclist frame, and the spacing between marks will be l, hence the above result.

Last edited: Sep 29, 2014
6. Sep 30, 2014

### A.T.

Here some
useful visualizations on this:

The key is that the lower part of the wheel doesn't move relative to the surface, so it's not length contracted in the surface frame. In the frame of the wheel's center, both the
surface and the lower wheel part are moving at the same speed, so they are contracted by the same amount.

So the gap between the marks that the spokes would make on the surface (measured in the surface frame) is constant (independent of the bikes speed)

7. Sep 30, 2014

### Jaams

I'm sorry for not doing a proper response to what you've said, but could the issues with the rotation or motion in general be pushed aside for a moment? In my mind the question and the inherent problem with the answer are more basic in nature.

I'm mostly interested in talking about the scenario purely from the perspective of length contraction, which says that his footprints should end up being further apart.. right? Yet an onlooker would say that the man taking longer steps is length contracted? This is where I'm stuck.

8. Sep 30, 2014

### Staff: Mentor

Every observer, no matter their state of motion and acceleration and reference frame, will agree about the number of times that a heel strikes the ground during the walk (that is, the number of strides). They will generally have different notions of which lengths are contracted by how much, and hence of what what the distance covered is and what the stride length is, but they will agree about the number of strides and that the stride length times the number of strides is equal to the distance covered.

9. Sep 30, 2014

### pervect

Staff Emeritus
Well, maybe the best idea to avoid rotation (a good plan, by the way) will be to use the pole-barn paradox or some variant thereof.

See for instance http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/polebarn.html for a detailed analysis of the classic problem which requires some familiarity with the Lorentz transform to understand.

The highlights:

The runner carries a pole along with them, of a known length. In the hyperphysics example, the pole is 20 meters long. There is a barn that is only 10 meters long. From the viewpoint of the barn, the pole Lorentz contracts and fits into the barn in such a manner that both doors can close with the pole inside it. From the viewpoint of the pole, the barn is length contratced, and the barn doors are never both closed at the same time.

The key points to understanding the paradox are:

1) Simultaneity is relative, and the notion of simultaneity affects how we measure the length of a moving object
2) Length is not independent of the observer in SR. The Lorentz interval is the only observer indpendent invariant, length, which used to be an invariant, is not an invariant.

The modified version closest to your original would be something like the following. A runner carries a fixed length pole, and either via a series of gates, or by drawing chalk marks on the ground, determines how many poles fit between the start and finish line. You could even have multiple runners carrying multiple poles.

The ground observer notes that the spacing between marks is not the same (longer) than the spacing in the runner's reference frame. WHen he pays attention to the simultaneity issue, he finds they aren't simultaneous, either.

10. Sep 30, 2014

### sweet springs

Hi.
Once he gets on a "high speed" IFR of his walking, outer space contracts. He can reach far away galaxy in less than a short time of one step.

11. Oct 1, 2014

### Simon Bridge

The way this usually goes is that some sort of "paradox" is being set up like this:
The traveller leaves footprints in the ground.
Since the traveller is moving, his stride is shorter (length contraction) and so you get more footprints between the start and finish of the journey - but - in the travellers frame, it is the distance from start to finish that is contracted, and so it takes fewer strides, so there should be fewer footprints.

Is this what is intended here?

12. Oct 1, 2014

### sweet springs

Hi.

Say original IRF is the IRF where he stayed still before walking.
Now he is walking near light speed to original IRF. In original IRF, the walker is thinned in direction of front-back by Lorentz contraction. His shoe size is shortend keeping the same width.
He is very slow in walking motion like lifting thigh up and lowering feet. Say tau is proper time for his one stride, it takes
$$\frac{\tau}{\sqrt{1-\beta^2}}$$
During the time he moves the distance
$$\frac{c\tau\beta}{\sqrt{1-\beta^2}}$$
It reaches to infinity when beta = v/c approaches 1.

With less than one step he can go infinitely beyond. He can go infinite forward with no foot steps.
Same result in different IRFs, no paradox.

Running needs quick exercise of body than walking. We tend to think near light speed requires quick motion of body with friction of the Earth to shoes, wind against or so in mind.
But I do not think this relation between speed and walking pace holds here. Kicking the Earth or moving feet in the air does not matter. For example jogging astronaut in the near light speed rocket is included in our case.

Last edited: Oct 1, 2014
13. Oct 1, 2014

### Simon Bridge

Before looking at ways to resolve the apparent paradox, I think we should make sure that this is what OP intended to talk about. Otherwise we risk hijacking the thread.

14. Oct 1, 2014

### sweet springs

You are right. I am interested in OP's setting of the proper pace, tau in my post.

15. Oct 1, 2014

### ghwellsjr

I voted yes. Also, the answer to your question in the first post is yes.

I think this can be adequately understood by drawing some spacetime diagrams.

In the rest frame of the man, I will assume that he has a 4-foot stride so that with each step when both feet are on the ground, one foot is 2 feet in front of him and the other foot is 2 feet behind him. Then the rear foot lifts off the ground and moves forward at some speed while the front foot moves backward at the same speed. The man remains half-way between his feet at all times and both feet are moving at the same speed but in opposite directions at all times. Thus, in his rest frame, the ground moves behind him at that same speed.

Here is a spacetime diagram depicting the above description where the man's feet are moving at 80%c where c is 1 foot per nanosecond. The man is shown in black, his right foot in red and his left foot in blue. The dots represent one-nanosecond increments of Proper Time for the man and his two feet:

Since the man has taken 6 steps covering 4 feet each, the ground has moved 24 feet behind him in 30 nanoseconds confirming a speed of 80%c.

Now we use the Lorentz Transformation process to see what this same scenario looks like in the rest frame of the ground:

Now we see that the man has traversed 40 feet in 50 nanoseconds, confirming the same speed of 80%c. However, his feet alternate between being at rest for about 3 nanoseconds and traveling forward at 97.561%c for about 13.5 nanoseconds. Their average stride is 6.67 feet so during this scenario of 6 steps they cover 40 feet. Since his average stride has increased from 4 feet to 6.67 feet, the answer to your vote is yes.

Note also that if the man had traversed the 40 foot distance at a normal slow speed with the same stride of 4 feet, it would have taken him 10 steps instead of 6, making the answer to your question in the first post yes.

16. Oct 1, 2014

### A.T.

The key is, that not all of the man is length contracted in the ground frame. When both feet are on the ground, they are at rest in the ground frame, so the distance between them in the ground frame is greater than in the man's frame.

If the legs are straight in the man's frame, they are bend in the ground frame, similar to the spokes of the
relativistic

17. Oct 1, 2014

### jartsa

Since the man has taken 3 steps over the ground with his left foot, left foot has moved 12 feet over the ground.

Man must stay near his left foot, which is covering the ground at speed 40%c, that's why man covers ground at speed 40%c.

The fact that there is also a right foot does not matter.

A conclusion from this: 50%c is the maximum walking speed.

Last edited: Oct 1, 2014
18. Oct 1, 2014

### ghwellsjr

No, that's not right. The man can walk at any speed short of c. Look at the second diagram. The man's body is traveling at 80%c. His left foot (blue) travels at 97.561%c but for about 13.5/16.5 % of the time so it comes out to 80%c.

19. Oct 1, 2014

### jartsa

I disagree. When walking, invariant time schedule of a foot is: half of the time on the ground, half of the time in the air.

An example of invariance of time schedules: I study physics 1 hour a day, astrology 3 hours a day, the ratio 1/3 is a frame independent invariant.

When foot is in the air, it is moving over the ground. So we have: half of the time foot is moving over the ground, half of the time it is not moving over the ground. Those pauses cause a 50% decrease of average speed.

(I have always thought that when I walk 1 km, I take 1000 one meter steps. But actually I take 1000 two meter steps, 500 with each foot.)

Last edited: Oct 1, 2014
20. Oct 1, 2014

### ghwellsjr

True, as long as you're talking about the invariant Proper Time as depicted by the blue and red dots which is 3 nanoseconds on the ground and 3 nanoseconds in the air moving forward to the right as both my diagrams indicate.

Yes, as long as all parts of your body are at rest with respect to each other. That's not the case with the walking man.

Half the time the left foot is pushing the ground behind the man and the other half the right foot is pushing the ground according to the man's rest frame. Look at the first diagram.

In the first diagram, the man is "taking a 24-foot walk" with six 4-foot steps, half with each foot.

21. Oct 1, 2014

### Jaams

I've tried to understand this scenario in a visual sense, but I don't know how to because I find poles and chalk markings to be too open for inaccurate interpretations. How does the onlooker notice that the spacing isn't the same (edit: How does it all add up)?

I'd also like to hear your opinion on this:
sweet springs: He can reach far away galaxy in less than a short time of one step.
If that's how length contraction works, how can an onlooker possibly see it as length contraction?

My goal is to be able to accurately understand and possibly explain relativity in the context of walking, so all approaches to this scenario are totally welcome.

22. Oct 2, 2014

### Simon Bridge

Often left out of relativity scenarios is how the observers find out that there is anything different.
It is simple enough to add in to the description - the method used will depend on the specifics of the question being asked.
Usually two observers take records of their measurements and meet up later to compare notes - or a number of experiments are conducted and the results compared.

The process of walking is quite complicated, so it is not a useful way to make the details of relativity clear. It seems you don't have a clear question - which means that we are unlikely to be able to help you better than just working through a relativity text. You have seen a number of approaches above that will provide starting points.

What happens is that the two observers see different sequences of events.
The ground-frame observer must see the walker taking smaller steps, we reason, because his stride is length contracted ... certainly if he was carrying a stick the same length as his stride, that stick will be shorter in the ground-frame.

It's similar to the old saw about the car speeding into a shed whose rest length is the same as the rest length of the car.
An observer in the frame of the shed realizes that the car will fit comfortably because it's length is contracted to smaller than the shed ... but an observer in the frame of the car sees the shed as smaller so what gives?

23. Oct 2, 2014

### jartsa

Yes, you are right. I must take the viewpoint of the man, who must take the viewpoint that the ground moving like treadmill surface.

And you are right about all the the other things too. I got a little bit confused.

24. Oct 2, 2014

### ghwellsjr

Well let's see if we can get you unstuck.

I've added some markings to my previous diagrams to explain what is Length Contracted in your scenario. Here's the first diagram:

In the man's rest frame his stride is 4 feet as marked off by the green lines.

Now we go to the rest frame of the ground:

Now the question is: what is Length Contracted in this frame? Certainly not the separation of the footprints--they are farther apart. In Special Relativity distances must be measured at the same time in any frame. We can see that the footprints are not placed at the same time--they are placed more than 8 nanoseconds apart. But if we look at the separation of the two green lines that represent the stride from the man's rest frame, we see that it is Length Contracted and by the exact amount. Since gamma at 80%c is 1.667 we divide that into the separation of the green lines from the man's rest frame and get 2.4 feet.

Now what about the separation of the footprints? Well, after they are laid down, they have a constant separation in the ground frame of 6.67 feet so it doesn't matter when we measure them but if we go back to the first diagram we have to consider the distance between two adjacent blue and red lines that are diagonally going up and to the left. Fortunately, they are coincident in time at the ending of one and the beginning of the other as they fall on the green lines and so we can show that the separation of the footprints from the ground rest frame of 6.67 is Length Contracted to the correct amount if we divide that by gamma and get 4 feet.

Length Contraction is defined in Special Relativity as the ratio of the endpoints of something moving in one frame and measured at the same time to its length in its rest frame. Hopefully, you can see how this definition is applied for both the man's stride and for the footprints on the ground.

25. Oct 2, 2014

### Simon Bridge

In other words - the ground observer sees the man taking long leaps - allowing him to cover more ground despite his short stride.
I like the diagrams BTW.

People not used to them can have trouble reading them though.