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.