Can the same ruler have two different sizes?

[geebor]

TL;DR

two light sources approaching at different speed require two different ruler sizes for both to be at the speed of light

I understand how my ruler size can change with my speed toward a light source so that it always appears to be approaching at the speed of light but what if there are two light sources approaching at different speeds. How can my same ruler be two different sizes at the same time?

 

[robphy]

While light travels at the speed of light,
“light sources” (e.g., light bulbs) don’t travel at the speed of light.

The answer to your last question is best answered by drawing a spacetime diagram of a ruler…and learning how an observer measures a length of that ruler.

 

[Nugatory]

I understand how my ruler size can change with my speed toward a light source so that it always appears to be approaching at the speed of light

I’m not sure what you’re trying to say here. Light sources, whether you’re moving towards them or not, have nothing to do with length contraction. The principle is less complicated: you will find a ruler that is moving relative to you to be shorter than if it were at rest relative to you. (This effect is symmetrical - someone moving relative to you will consider your ruler to be the moving one, hence length contracted).

The easiest way to see how this can be is to look at a Minkowski diagram showing the path through spacetime of the two ends of the ruler. The length of the ruler is of course the distance between where the ends are at the same time.

If you are not already familiar with the relativity of simultaneity (Google, or many good threads here) it is essential that you learn about it before you dig any deeper into length contraction and time dilation.

 

[FactChecker]

The relativity of simultaneity is the key. In order to measure the length of a moving ruler, you must measure the location of both ends at the same time from your perspective. Two people in different inertial reference frames (IRF) will not agree on when position measurements of the ends are at the same time. If someone in a different IRF measures the position of the leading end of the ruler before (according to you) measuring the trailing end position, then you will think that his length calculation is too short. But he will think he did it right. So the same ruler will have a different length for each different IRF.

 

[Ibix]

TL;DR Summary: two light sources approaching at different speed require two different ruler sizes for both to be at the speed of light

I understand how my ruler size can change with my speed toward a light source so that it always appears to be approaching at the speed of light but what if there are two light sources approaching at different speeds. How can my same ruler be two different sizes at the same time?

As others have commented this is a difficult to parse, but are you imagining that light from different sources is doing different speeds and only length contraction causes light to have a measured speed of $c$? If so, this is incorrect. The speed of light in vacuum is $c$ independent of its source.

Where length contraction, time dilation, and (importantly) the relativity of simultaneity come in is in understanding how another observer in motion with respect to you can also measure $c$ for the speed of the same pulse of light that you have measured to be travelling at $c$.

 

[geebor]

OK I didn't pose the question correctly. Supposing I am between two light sources L1 and L2. I am moving toward L2 and away from L1. I measure the speed of light from L1 and the speed of light from L2. My ruler and my clock change with my speed so that the faster I approach L2 the larger my ruler is so that it appears that light from L2 is going always at speed C. My question is the speed at which I approach L2 is the same as the speed that I leave L1. In order to measure the speed of light from L1 as C my ruler has to contract as my speed increases. But my ruler is expanding to measure the speed of light from L2 as C. How can my ruler expand and contract at the same time?

 

[FactChecker]

First of all, if you are an observer in an inertial reference frame then you are perfectly justified in assuming that you are stationary and nothing needs to be explained to you.
Your question is how an outside observer, who thinks that you are moving, can rationalize that you measure both light pulses at the same speed of c while using one ruler. You are overlooking the relativity of simultaneity. You should consider this as an outside observer would. The outside observer thinks that your clocks are set wrong. He thinks that your clock at the leading end of the ruler is set behind when compared with your clock at the trailing end of the ruler. When you time the light pulse coming at you from the front, the light pulse passes the leading end clock first and later passes the trailing clock. But the leading end clock was set behind, so you calculate a slower relative light speed than you should. On the other hand, when you time the light pulse coming at you from the rear, the light pulse passes the trailing end clock first and later passes the leading clock. But the leading end clock was set behind, so you calculate a faster relative light speed than you should. In both light pulses, you calculate relative speeds of exactly c. So an outside observer understands that you can use one ruler to get a speed of c for both light pulses because you use clocks that he thinks are set wrong.

 

[geebor]

Thanks for clearing that up.

 

[Ibix]

As a general rule, @geebor, do not try to reason from length contraction and/or time dilation. They are both special cases and leave out the relativity of simultaneity, which is the de-synchronised clocks effect that @FactChecker explained above. Always write down the coordinates of events of interest and use the full Lorentz transforms to carry out the analysis.

Once you are confident with those and have built up some intuition, then you will develop a reasonable idea of when it is safe can short-cut the analysis with length contraction or time dilation. Not realising that you are in a situation where you can't take that short-cut is one of the most common sources of confusion in people learning relativity.

 

[FactChecker]

Always write down the coordinates of events of interest and use the full Lorentz transforms to carry out the analysis.

Yes! Trying to keep track of all the things that change will drive a person crazy. The full Lorentz transformations and tools like Minkowski spacetime diagrams are great for that. When trying to understand SR, protect your sanity by using those tools. ;-)

 

[robphy]

Always write down the coordinates of events of interest and use the full Lorentz transforms to carry out the analysis.

A full Lorentz transform isn’t always necessary. Often, a spacetime-trigonometry analysis on a spacetime diagram is sufficient… but requires practice.

(Analogously, a full Euclidean rotation [and translation] isn’t always necessary for a plane geometry problem. Often, a trigonometry analysis is sufficient.)

 

[PeterDonis]

How can my ruler expand and contract at the same time?

Nothing is happening to your ruler. Other observers, moving at different speeds relative to you, measure your ruler to be length contracted by different amounts (nobody sees it "expanded" compared to its rest length, the length it is to you), because they are viewing your ruler from different "angles" in spacetime.

 

[PeterDonis]

In order to measure the speed of light from L1 as C my ruler has to contract as my speed increases. But my ruler is expanding to measure the speed of light from L2 as C.

Neither of these are correct for you; to you your ruler remains at its rest length. L1 is moving away from you and L2 is moving towards you, and all you have to do is take into account the change in distance due to their motion, as measured by your ruler, and the change in light travel time from each of them, to confirm that light from both of them moves at $c$ relative to you.

 

[FactChecker]

But my ruler is expanding to measure the speed of light from L2 as C. How can my ruler expand and contract at the same time?

An outside observer always thinks that a ruler you say is 1 ft long is actually shorter than 1 ft. This is true no matter which direction you are going. Again, it is the relativity of simultaneity at work. To measure a moving ruler, you must measure the position of both ends at the same time. But the outside observer thinks that you have set your clocks wrong. The memory trick is "Ahead is behind and behind is ahead." Meaning that the outside observer thinks that your clock at the leading end of the ruler is set behind in time compared to your clock at the trailing end (according to the outsider's clocks). This is true no matter what direction you are going. When the outside observer sees you measure the length of what you think is a 1 ft ruler, he sees you measure the trailing end position, then you wait a little while as you and the ruler go forward and finally measure the position of the leading end. So he believes that you have stretched the length forward when you said it is one ft. That is, he believes that the ruler is less than one foot. And if he measures the same ruler using his clocks to "simultaneously" position both ends, he does get a shorter measurement.

 

[obtronix]

TL;DR Summary: two light sources approaching at different speed require two different ruler sizes for both to be at the speed of light

I understand how my ruler size can change with my speed toward a light source so that it always appears to be approaching at the speed of light but what if there are two light sources approaching at different speeds. How can my same ruler be two different sizes at the same time?

From someone else's inertial reference frame perspective there are an infinite number of possible ruler sizes, this is no different than me measuring a tree at different angles or at different distances and measuring different tree heights. In special relativity spacetime rotation is due to relative velocity. It's all geometry.