Does the Lorentz factor aid understanding of SR, or obscure it?

[Trysse]

I had a look at a number of books that deal with Special Relativity.

Many, if not most, textbooks on the theory of Special Relativity introduce the Lorentz factor $\gamma$
$$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
Different textbooks introduce $\gamma$ differently; some use it only as a shorthand to replace the mathematical expression above. Some introduce it with a historic reference to Hendrik Lorentz. And others again dive more deeply into how $\gamma$ relates to relative speed.

However, some textbooks do not introduce $\gamma$ as a distinct symbol. In these books the authors simply use $\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ or $\sqrt{1-\frac{v^2}{c^2}}$ wherever needed.

And of course $\gamma$ pops up in many discussions here on PF.

So would like to know what you think:

If you were to write a textbook, would you use $\gamma$ or not? If yes, how? If you teach SR, do you use $\gamma$? Do you think $\gamma$ was helpful when you learned SR?

To me, the benefit of $\gamma$ is that almost everybody knows it. So when I say $\gamma$ everybody knows what I mean. I don't need to explain it. But I don't think it's helpful for understanding Special Relativity. I actually think it hides how special relativity works.

Do you have an opinion? Or is this a topic you have not yet thought about?

 

[weirdoguy]

I actually think it hides how special relativity works.

How? It's just a notation. Writing down this square root explicitly every single time is tiring. That's all to this.

EDIT: You've had similar thread about this: https://www.physicsforums.com/threads/lorentz-factor-in-textbooks-on-special-relativity.1078774/

 

[vela]

I use it mainly for the same reason @weirdoguy stated. It's a pain to keep writing the square root down, and not using it makes the equations look unnecessarily complicated. Another reason is that I find my students are notoriously bad at calculating things. If they can calculate $\gamma$ just once and use that value elsewhere, I'd wager it results in fewer errors than if they had to evaluate the square root in every single equation.

I'm not sure how using $\gamma$ hides how SR works. I think the best way to see how SR works is to write the equations in terms of rapidity.

 

[sbrothy]

Isn't it at the very core of SR? That velocities don't add but rapidities do?

How can you talk about time dilation and not reaching light-speed without the gamma-factor?

 

[Trysse]

How can you talk about time dilation and not reaching light-speed without the gamma-factor?

I am not sure if this was meant as a rhetorical question. But I think this is the right question to ask. I would reformulate it slightly:

Is the Lorentz factor i.e. the expression
$$\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
fundamentally necessary to do the math of Special Relativity?

 

[sbrothy]

I think so yes. I'm not as strong in math but I think that was essentially what I meant. I can't remember the particular equation but it goes something like this:

γ = 1 / Sqrt[ 1 - (v2 / c2) ]

which looks suspiciously like yours!

EDIT: I'm just not so strong in Latex.

EDIT: But thinking about who I'm "discussing" with you're probably right. It was, in all likelihood, rhetorical.

 

[weirdoguy]

I am not sure if this was meant as a rhetorical question. But I think this is the right question to ask.

And the answer is yes, whether you write gamma, or write square root explicitly. You can't avoid it.

 

[robphy]

I am not sure if this was meant as a rhetorical question. But I think this is the right question to ask. I would reformulate it slightly:

Is the Lorentz factor i.e. the expression
$$\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
fundamentally necessary to do the math of Special Relativity?

The quantity it represents is important, but not necessarily expressed in terms of the spatial velocity $v$.

I use it mainly for the same reason @weirdoguy stated. It's a pain to keep writing the square root down, and not using it makes the equations look unnecessarily complicated. Another reason is that I find my students are notoriously bad at calculating things. If they can calculate $\gamma$ just once and use that value elsewhere, I'd wager it results in fewer errors than if they had to evaluate the square root in every single equation.

I'm not sure how using $\gamma$ hides how SR works. I think the best way to see how SR works is to write the equations in terms of rapidity.

Indeed, $$\gamma=\frac{1}{\sqrt{1-(v/c)^2}}=\cosh\theta,$$ where $\theta$ is the rapidity (the Minkowski-analogue of the angle between two vectors)...
and $\beta=(v/c)=\tanh\theta$ and $k=\sqrt{\frac{1+(v/c)}{1-(v/c)}}=\exp\theta$.
In Euclidean geometry, we usually think in terms of angles and trig-functions, as opposed to slopes, like $m$.
$$\cos\phi\equiv\frac{1}{\sqrt{1+\tan^2\phi}}=\frac{1}{\sqrt{1+m^2}}.$$

---

I would argue that special relativity would be easier to understand
if we made use of the trigonometry one learned in PHY 101,
suitably modified to the hyperbolic-trigonometry needed for special relativity.

Many textbook problems in special relativity are the analogues of trigonometry problems in Euclidean geometry. While there is still some non-intuitive physics to understand, I think that a hyperbolic-trigonometric method of computation makes the calculations easier to perform and trust and gradually interpret (as compared to the unfamiliar algebra required when working with $v$ and $\gamma$).

As a possible starting point, given two 4-velocities (future-pointing unit timelike 4-vectors),
their Minkowski-dot-product is the time-dilation factor:
$$\gamma=\hat u \stackrel{M}{\cdot} \hat t=\cosh\theta,$$ where $\theta$ is the relative-rapidity between the two observers.
(Then $\tanh\theta$ is the relative-velocity.)

One can work with Minkowski-right triangles and interpret
$$\gamma=\cosh\theta=\frac{\rm ADJACENT}{\rm HYPOTENUSE}$$ and reason geometrically [with some guidance on what Euclidean ideas persist
and what Euclidean ideas need to be generalized for "spacetime geometry" and "spacetime trigonometry"].
Time-dilation is spacetime-geometrically the projection of the hypotenuse (representing the "traveler's elapsed proper time" onto the adjacent side of the measuring-observer.

Slide the E-slider from +1 to -1 to see the Euclidean version.
desmos.com/calculator/hjqwqjlr6k

 

[Herman Trivilino]

I start with a chart showing the values of $v$ and $\gamma$ for the motions of different objects. Starting with
maybe a race car, then a bullet, and so on up to the speed of Earth orbiting the sun, showing that what we think of as everyday speeds have a value of $\gamma \approx 1$. Then I move on to particles in say, proton therapy, then research particle accelerators.

From there equations such as $\Delta t = \gamma \Delta \tau$ or $E=\gamma mc^2$ are more meaningful and easier to read. This also emphasizes how very fast speeds must be for relativistic effects to be significant.

 

[PeroK]

I start with a chart showing the values of $v$ and $\gamma$ for the motions of different objects. Starting with
maybe a race car, then a bullet, and so on up to the speed of Earth orbiting the sun, showing that what we think of as everyday speeds have a value of $\gamma \approx 1$. Then I move on to particles in say, proton therapy, then research particle accelerators.

From there equations such as $\Delta t = \gamma \Delta \tau$ or $E=\gamma mc^2$ are more meaningful and easier to read. This also emphasizes how very fast speeds must be for relativistic effects to be significant.

For most students, including myself, putting SR in this context is important. I'm quite mathematical, but diving straight into hyperbolic geometry is too abstract for me. That can come later.

 

[robphy]

For most students, including myself, putting SR in this context is important. I'm quite mathematical, but diving straight into hyperbolic geometry is too abstract for me. That can come later.

Note: special-relativity is hyperbolic-trigonometry in flat Minkowski spacetime
(not curved hyperbolic geometry, unless you are talking about the velocity-space or the mass-shell).

Euclidean geometry is circular-trigonometry in flat Euclidean space
(not curved spherical geometry, unless you are talking about the space of directions [in the sky]).

---

What I advocate is

• generalizing position-vs-time diagrams (aka spacetime diagrams) that one learned in PHY 101,
  not space-diagrams of moving box-cars.
• generalizing trigonometric ideas that one learned from vectors and free-body diagrams in PHY 101,
  emphasizing ratios of triangle sides (and not as much about rapidity-measurements).

It seems the typical intro textbook treatments of special relativity don't make use of these skills
(often deemed too mathematical, as Einstein first declared after Minkowski's spacetime formulation).
So, instead, textbooks follow Einstein's original non-geometrical formulation and various thought-experiments, non-intuitive effects, and so-called-paradoxes.
... a method different from how Galilean physics is taught.
I think using some geometrical ideas alongside the typical textbook treatment would help... but the instructor has to be willing and adequately prepared.

 

[sbrothy]

[...]

I'm not sure how using $\gamma$ hides how SR works. I think the best way to see how SR works is to write the equations in terms of rapidity.

I'm not at all sure it does. Quite the opposite in fact. But you're right. Understanding the field equations and how they fit together is probably the most important lesson.

EDIT: I've met people who thought that each of Einsteins equations had it own function within the framework. Not so long ago someone on here asked if the solution to the EFE's could really be 12(?!)

EDIT2: I wish it were that simple!

 

[sbrothy]

For most students, including myself, putting SR in this context is important. I'm quite mathematical, but diving straight into hyperbolic geometry is too abstract for me. That can come later.

I've tried to read Penrose's "Road to Reality" and he brings up the hyperbolic geometry very early.

EDIT: Then again he does the same with Riemannian geometry. I suspect that man doesn't suffer fools gladly.

 

[robphy]

I'm not at all sure it does. Quite the opposite in fact. But you're right. Understanding the field equations and how they fit together is probably the most important lesson.

EDIT: I've met people who thought that each of Einsteins equations had it own function within the framework. Not so long ago someone on here asked if the solution to the EFE's could really be 12(?!)

EDIT2: I wish it were that simple!

This thread is about Special Relativity, where the field equations are not treated.

I've tried to read Penrose's "Road to Reality" and he brings up the hyperbolic geometry very early.

EDIT: Then again he does the same with Riemannian geometry. I suspect that man doesn't suffer fools gladly.

For an introduction to Special Relativity, I would not recommend Penrose.
Hyperbolic-geometry (a curved 3-D space) occurs in certain subspaces (like the space of 4-velocities) in flat 4D-Minkowski spacetime... and this is an advanced topic (akin to spherical geometry to study the surface of the Earth for navigation).

Hyperbolic-trigonometry, by contrast, is involved with the flat 4D Minkowski spacetime (and in particular with the familiar (1+1)-D spacetime diagram. This is akin to circular-trigonometry which is involved with the flat Euclidean geometry.

It is a common error to confuse
hyperbolic-trigonometry (for a flat spacetime) with hyperbolic-geometry (a curved space).
By analogy, we have
circular-trigonometry (for a flat space), which we shouldn't confuse with spherical-geometry (a curved space).

 

[sbrothy]

Yeah, but as I said I'm an autodidact hack who often get's in over his head. It was an offhand remark not really connected with the topic. More bad habits I have to get rid of. I apologize.

EDIT: I learn something every time I'm put in my place though.

EDIT2: Much like lifting a cat by the tail. You learn something you can't learn any other way.

 

[Trysse]

Sorry that it took me so long to come back to this thread. I was waiting to see if there was somebody who shares my issues with the Lorentz factor $\gamma$. And also, I took me a while to consider my responses.

I'm not sure how using γ hides how SR works.

I was intentionally ambiguous here as I was wondering if others would come up with similar issues. However, since nobody chimed in, I might as well make my issues explicit and pick up my own thread.

It's just a notation.

I agree insofar as, for actual calculation, it makes little difference how I write an equation or mathematical expression, as long as the results are correct. However, I believe there are some notations that are better than others.
Good notation makes clear what is happening. It puts the most important objects at the centre. It groups the objects that are somehow related close together. If physical units are involved, the units are grouped in such a way that it easily shows how the units are related and/or cancel out where possible. The notation should also make it easy to see quantitative relations between the different elemets/parameters. Good notation does not introduce unnecessary calculation steps or objects.
Just because a notation is widely used (such as $\gamma$ does not necessarily mean it is a good notation. Or at least it does not forbid looking at the notation critically. This is what I invite you to do.
I think $\gamma$ checks at least some points where it is not best in class.

I think the best way to see how SR works is to write the equations in terms of rapidity.

I cannot comment on this approach as I have not yet thought about it in any depth. So I will not respond to this argument. Not because I think it does not make a point, but I think it is not relevant to the issues I have with $\gamma$, and I have no opinion. If you discuss this in another thread and other members will discuss, I will surely follow.

Another reason is that I find my students are notoriously bad at calculating things. If they can calculate γ just once and use that value elsewhere, I'd wager it results in fewer errors than if they had to evaluate the square root in every single equation.

I can agree with the argument that calculations which are repeated over and over again should be done once, and the intermediate result can be reused. But I think the result this calculation gives is not as meaningful as is sometimes claimed. See also my response to @Herman Trivilino below.

I'm not as strong in math

I find this is a good answer on why people might think $\gamma$ is indispensable: $\gamma$ is used everywhere. It must be indispensable even if I do not understand what it does or how it does it.
But also people who are strong in math might have the same impression for a similar but slightly different reason: They use $\gamma$ so much that for them it has become indispensable.

And the answer is yes, whether you write gamma, or write square root explicitly. You can't avoid it.

Although I disagree with the yes, I think this makes the right point. Of course, I can avoid
$$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
What I cannot avoid is the "square root" term in the denominator of $\gamma$
$$\gamma^{-1}=\sqrt{1-\frac{v^2}{c^2}}$$.

What I can do: Whenever I am supposed to multiply with $\gamma$ I can also divide by $\gamma^{-1}$. And of course, vice versa for division.

And this is already my main issue with the Lorentz factor $\gamma$: It "hides" the expression that does all the heavy lifting $\sqrt{1-(v/c)^2}$ in the denominator of an unnecessary fraction. It is unnecessary because the nominator is a $1$ that has no mathematical or physical meaning. Its only function is to create an expression which can be written as a factor in the Lorentz transformation. However, the expression does not easily fit in with mathematical and physical intuition.

That this expression is odd is not only my personal feeling. Here is a quote from Spacetime Physics: Introduction to Special Relativity by Taylor and Wheeler, (1992):

"The awkward expression $\frac{ 1}{\sqrt{1-\frac{v^2}{c^2}}}$ occurs often in what follows.
For simplicity, this expression is given the symbol Greek lower-case $\gamma$."

I start with a chart showing the values of v and γ for the motions of different objects. Starting with
maybe a race car, then a bullet, and so on up to the speed of Earth orbiting the sun, showing that what we think of as everyday speeds have a value of γ≈1. Then I move on to particles in say, proton therapy, then research particle accelerators.

This use of $\gamma$ shows students why they do not experience relativistic effects at everyday speeds. And for this purpose, it might be helpful and is certainly well established. But I think it does not make it easy to see why relativistic effects relate to relative speed the way they do.
Analysing $\gamma^{-1}$ offers a much better intuition on why relativistic effects behave the way they do.

The quantity it represents is important,

I find any quantity that grows without bounds while the input quantity/parameter (i.e. $v$) is finite suspicious. Again, I think $\gamma^{-1}$ offers a better (i.e. bounded) alternative.

For now, I will not go into more details on why I think $\gamma^{-1}$ is simpler and at the same time more explanatory than $\gamma$. For once, I think this post is already long, and secondly, I think it might be more fun if you stare at
$$\gamma^{-1}=\sqrt{1-\frac{v^2}{c^2}}$$
for a while and think what this term can tell you about relativistic effects, that $\gamma$ does not reveal as easily.
Although I prefer
$$\gamma^{-1}=\sqrt{1-\beta^2}$$
with
$$\beta=\frac{v}{c}$$

As an anecdote: The foundation for this thread goes back to 2021 when @robphy helped me to fix a 2+1 Minkowski diagram of the Michelson and Morley experiment. He pointed out that I must take into account $\gamma$ for length contraction. As I proceeded to create several versions of the model, I found that it is simpler to write only the denominator of $\gamma$. However, back then, it did not occur to me that $\gamma$ might not be a fundamental and explanatory term/concept/quantity.

 

[renormalize]

This use of $\gamma$ shows students why they do not experience relativistic effects at everyday speeds. And for this purpose, it might be helpful and is certainly well established. But I think it does not make it easy to see why relativistic effects relate to relative speed the way they do.
Analysing $\gamma^{-1}$ offers a much better intuition on why relativistic effects behave the way they do.

I find any quantity that grows without bounds while the input quantity/parameter (i.e. $v$ ) is finite suspicious.

I disagree. Consider this right triangle:

We have the "awkward expression":$$\tan\theta\equiv\frac{c}{a}=\frac{1}{\sqrt{\frac{b^{2}}{c^{2}}-1}}$$And obviously $\tan\theta$ is a "quantity that grows without bounds" as $b\rightarrow c$. Do you therefore advocate avoiding the "suspicious" quantity $\tan\theta$ when doing ordinary trigonometry? If you don't, why should the hyperbolic trigonometry of special relativity be treated any differently?

 

[robphy]

For now, I will not go into more details on why I think $\gamma^{-1}$ is simpler and at the same time more explanatory than $\gamma$. For once, I think this post is already long, and secondly, I think it might be more fun if you stare at
$$\gamma^{-1}=\sqrt{1-\frac{v^2}{c^2}}$$
for a while and think what this term can tell you about relativistic effects, that $\gamma$ does not reveal as easily.
Although I prefer
$$\gamma^{-1}=\sqrt{1-\beta^2}$$
with
$$\beta=\frac{v}{c}$$

As an anecdote: The foundation for this thread goes back to 2021 when @robphy helped me to fix a 2+1 Minkowski diagram of the Michelson and Morley experiment. He pointed out that I must take into account $\gamma$ for length contraction. As I proceeded to create several versions of the model, I found that it is simpler to write only the denominator of $\gamma$. However, back then, it did not occur to me that $\gamma$ might not be a fundamental and explanatory term/concept/quantity.

As a student, and later as an instructor, I find it useful to draw connections and make appropriate analogies with previous knowledge and foreshadow connections with future developments--rather than treating things in isolation, possibly with a narrow perspective for a particular point of convenience.

Since "spacetime geometry" is a key concept in relativity, first developed by Minkowski and eventually appreciated by Einstein, it would be good to point out that promoting
$$\gamma^{-1}=\sqrt{1-\beta^2}$$
is analogous to promoting $$\sec\theta=\frac{\rm HYPOTENUSE}{\rm ADJACENT}$$ in the Euclidean case.

Since "length contraction" seems to be a particular point of focus for you, it would be good realize that
"length contraction" is the Minkowski-analogue of the https://en.wikipedia.org/wiki/Distance_between_two_parallel_lines
where, for slope $m$,
$$d=\frac{|b_2-b_1|}{\sqrt{m^2+1}}.$$
The factor $\frac{1}{\sqrt{m^2+1}}$ is $\cos\theta$, and thus $\sec\theta=\sqrt{m^2+1}$.

But, of course, since Euclidean geometry is more than just the distance between two lines. It would be good to consider how one would do Euclidean geometry with emphasis on $\sec\theta$.

From my own point of view, "length contraction" is merely an effect of relativity... but it's not a primary concept. I think "time dilation" is more important... but even that isn't primary.
To me, casuality and the light-cone (invariant finite speed of signal propagation) are more important. (In terms of "relativistic factors", the Doppler factor $k$ is more physically important... [mathematically, it's an eigenvalue of the eigenvectors of a Lorentz transformation].)