# Comparing rates -- Why don't they cancel?

• B
Gold Member
I know the surface of the Lorentz transformations.

When comparing two rates though, why is it simply not rate over rate (rate/rate)?

Like when comparing two rates of time, due to time dilation, why is it not just time over time (time/time)? I can see that the units would cancel to just 1, and so you'd have no such thing as time. Meters per second compared to different meters per second. (m/s)/(m/s)=unitless. What is the foundation for comparing these so the units don't cancel?

PeroK
Homework Helper
Gold Member
2020 Award
I know the surface of the Lorentz transformations.

When comparing two rates though, why is it simply not rate over rate (rate/rate)?

Like when comparing two rates of time, due to time dilation, why is it not just time over time (time/time)? I can see that the units would cancel to just 1, and so you'd have no such thing as time. Meters per second compared to different meters per second. (m/s)/(m/s)=unitless. What is the foundation for comparing these so the units don't cancel?
You'll have to explain what you mean by that.

• Ibix
2020 Award
If you are writing a ratio of two times, why would you expect the units not to cancel?

Gold Member
If you are writing a ratio of two times, why would you expect the units not to cancel?
My question is why is it not just a ratio?

Ibix
2020 Award
My question is why is it not just a ratio?
Why is what not just a ratio?

Gold Member
Why is what not just a ratio?
Ratios of time: s/s. Ratios of rates too like (m/s)/(m/s). You could even have rates of time compared: (s/s)/(s/s).

Ibix
2020 Award
Ratios of what times? At the moment, you're just saying "why is a ratio of times not a ratio", which is obviously self-contradictory.

• PeterDonis
Mentor
2020 Award
I know the surface of the Lorentz transformations.
I have no idea what you mean by this; the Lorentz transformations are not the kind of thing that has a "surface".

when comparing two rates of time, due to time dilation, why is it not just time over time (time/time)?
Please give a specific example of what you are describing here. A reference would help.

Mark44
Mentor
I know the surface of the Lorentz transformations.
I have no idea what you mean by this; the Lorentz transformations are not the kind of thing that has a "surface".
Maybe he means that he has a superficial understanding of Lorentz transformations; i.e., just at the surface level and no deeper.

• jaketodd
Gold Member
Ratios of what times? At the moment, you're just saying "why is a ratio of times not a ratio", which is obviously self-contradictory.
Let's say a fast moving object has time slowed down for it from your reference frame. When comparing the elapsing of time for that object, and for you, why is it not just s/s, which cancels to unitless? Hmm, maybe it would be s/s/s = s/s^2 = 1/s. Or (s/s)/(proper time), where proper time is some sort of absolute to compare all clocks to?

Comparing the speed of that object to yourself. Why is it not just (m/s)/(m/s), which cancels to unitless?

Maybe what I'm missing is subtraction. (m/s)-(m/s). And for time, s-s. For some reason I'm thinking division though. Comparing rates of time. (s/s)-(s/s) still cancels to no units.

• robphy
Homework Helper
Gold Member
Like when comparing two rates of time, due to time dilation, why is it not just time over time (time/time)? I can see that the units would cancel to just 1, and so you'd have no such thing as time.
The geometric analogue of time-dilation is that $\cos\theta= \frac{\mbox{adjacent side}}{\mbox{hypotenuse}}$, which is unitless.
The time-dilation factor (a ratio of two time intervals) is unitless.

In Euclidean geometry, would you say that
"since the units of length cancel to 1, there is no such thing as length?"

As others have suggested, maybe you need a clearer example.

• sysprog
Ibix
2020 Award
Let's say a fast moving object has time slowed down for it from your reference frame. When comparing the elapsing of time for that object, and for you, why is it not just s/s, which cancels to unitless? Hmm, maybe it would be s/s/s = s/s^2 = 1/s. Or (s/s)/(proper time), where proper time is some sort of absolute to compare all clocks to?

Comparing the speed of that object to yourself. Why is it not just (m/s)/(m/s), which cancels to unitless?

Maybe what I'm missing is subtraction. (m/s)-(m/s). And for time, s-s. For some reason I'm thinking division though. Comparing rates of time. (s/s)-(s/s) still cancels to no units.
Well, if you are comparing elapsed times by dividing one by the other the result will be dimensionless, as robphy has pointed out. The time between ticks of a clock moving at speed ##v## relative to you divided by the time between ticks of a clock at rest relative to you is ##\gamma=1/\sqrt{1-v^2/c^2}##, which is dimensionless.

I'm not seeing the problem.

• sysprog
PeterDonis
Mentor
2020 Award
When comparing the elapsing of time for that object, and for you, why is it not just s/s, which cancels to unitless?
How are you comparing the time? If you are just taking the ratio of the times, then it will be unitless, as @Ibix has pointed out.

I don't understand what you think the problem is.

I know the surface of the Lorentz transformations.

When comparing two rates though, why is it simply not rate over rate (rate/rate)?

Like when comparing two rates of time, due to time dilation, why is it not just time over time (time/time)? I can see that the units would cancel to just 1, and so you'd have no such thing as time. Meters per second compared to different meters per second. (m/s)/(m/s)=unitless. What is the foundation for comparing these so the units don't cancel?
Isn’t this just the Lorentz factor?

##γ = \frac{dt}{dτ}##

• PeroK
Gold Member
So what is the significance that there are unitless elements in the Lorentz Transformation?

Do they give any insight into what time is or what it isn't?

Maybe something in how those values are derived?

Does knowing how fast a clock ticks require an understanding of time apart from the clock - something to compare it to?

robphy
Homework Helper
Gold Member
So what is the significance that there are unitless elements in the Lorentz Transformation?
A Lorentz transformation is a type of linear transformation.
A rotation (a certain Euclidean transformation) is a similar type of linear transformation. Its elements are also unitless.
One reason they have to be unitless is that if I apply two rotations in the plane with two angles,
the result is equivalent to a single rotation using the sum of the angles.
So, the elements of the rotation have to be unitless.
This also applies to the Lorentz transformations (where the coordinates are normalized to be of the same type, e.g. (ct,x,y,z)).

Do they give any insight into what time is or what it isn't?
It suggests that coordinates of time and of space are inter-related by the symmetry
implied by the principle of relativity (in special relativity and in Galilean relativity).

So what is the significance that there are unitless elements in the Lorentz Transformation?
I think you are still trying to find here some mystery, but there is nothing mysterious about having dimensionless number after dividing two numbers with the same units.
Do they give any insight into what time is or what it isn't?
No.
Maybe something in how those values are derived?
How is the Lorentz transformation or time dilation derived? There are tons of material out there, textbooks, online lectures etc. Even wiki can provide you some insight.
Does knowing how fast a clock ticks require an understanding of time apart from the clock - something to compare it to?
Sure, if you want to claim that some clocks are running at slower rate, you must compare them with some other clocks. How else you could tell?

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• sysprog
Gold Member
Sure, if you want to claim that some clocks are running at slower rate, you must compare them with some other clocks. How else you could tell?
Seems there needs to be something absolute about time that enables the comparison of clocks.

There are parts to the human brain that keep time: https://www.sciencedaily.com/releases/2005/10/051028142649.htm

If we didn't have those parts of the brain, it seems we wouldn't be able to compare clocks. Same if computers didn't have clocks internally. It's interesting that we use these things to measure something we think is innate in the universe. Well the equations do yield correct results...

PeterDonis
Mentor
2020 Award
Seems there needs to be something absolute about time that enables the comparison of clocks.
There is: proper time. Which in relativity means the invariant arc length along a timelike worldline. "Comparison of clocks" just means comparing the arc lengths along different worldlines.

If we didn't have those parts of the brain, it seems we wouldn't be able to compare clocks. Same if computers didn't have clocks internally.
Not at all. Atomic clocks, quartz crystal clocks (like the one that's probably in your wristwatch), and many other kinds of clocks work just fine without any human brains or computers being involved.

It's interesting that we use these things to measure something we think is innate in the universe.
In relativity terms, as above, clocks are measuring the arc length along their worldlines. What's the problem?

• sysprog
Gold Member
Just funny the reliance on things to try to figure out The things. Like a little piece of brain tissue, ticking, and thereby drawing conclusions about the nature of all things.

If we didn't have those parts of the brain, it seems we wouldn't be able to compare clocks. Same if computers didn't have clocks internally. It's interesting that we use these things to measure something we think is innate in the universe. Well the equations do yield correct results...
"these things" are part of the universe, and obeying its laws, including the implications of special relativity. So why not?

The most simple clocks you could imagine are probably light clocks. I suggest you read this part of Feynman lecture. The mechanism of other clocks like for example human body, quartz or atomic clocks, is generally more complicated. But all of them can be (and really are) used to measure the time.

Side note:
I do not recommend to read further parts of the suggested lecture, beginning by 15–8 Relativistic dynamics. Feynman used a concept of "relativistic mass" which is now depreciated.

Gold Member
Seems like a scavenger hunt. We are not provided with the laws. No we have to go out using ourselves as tools to learn the laws. I guess it's more exciting that way.

Mister T
Gold Member
Meters per second compared to different meters per second. (m/s)/(m/s)=unitless. What is the foundation for comparing these so the units don't cancel?
You can do it either way. The choice is arbitrary. You can say that the speed of planet Earth in its orbit about the sun is 30 km/s. Or you can compare it to the speed of light, 300 000 km/s, and say that its speed is one ten-thousandth the speed of light.

Every measurement is a comparison to a standard, so this can be done with any measurement.

Gold Member
Someone says clock A ticks twice as fast as clock B. So 5 seconds of clock B is 10 seconds on clock A. So it could be said that (2 x clock B) / (1 x clock A). The unit of seconds would cancel. I think you guys already answered this as I am thinking of a ratio, that is normally unitless? Just seems like a rate of time would involve division and the units would cancel, so how is there any comparing time?

PeterDonis
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2020 Award