Solving Special Relativity Problems: Interval vs Lorenz Equations

[arydberg]

Why is it that in SR we always seem to jump to the Lorenz equations when there is a simpler way. This is the concept of an interval. The interval is defined as the square root of ( T squared minus X squared) . In Special relativity the time and distance are different for different observers but the interval is always conserved.

For instance here is a relativity problem solved by both methods. The units used are feet, and nanoseconds. With these units the velocity of light is equal to 1 ( with a 2% error) G = gamma or 1/square root (1-V^2/C^2)

Problem:

X and T are the platform coordinates and X' and T' are moving train coordinates.

A speeding train passes a 500 foot station platform. It's velocity is equal to 0.6 . At the entry end of the platform X = 0 and T = 0 . and T' = 0 . How old is the train engineer as the train passes the exit end of the platform.

The time to transit platform = T = D/V = 500/ .6 = 833.3333 nanoseconds

Lorenz equations:

X' = G * ( X - V*T) ( not used)

T' = G * ( T - X*V )

G = 1.25 ( for V = .6 )

now T' = 1.25 * ( 833.333 - 500 * .6) = 666.6666 nanoseconds for the age of the engineer

Interval method :

By The interval method the interval on the platform between the train entering one end and exiting the other end is square root ( T ^ 2 - X^2 ) or I = square root (833.333 ^2 - 500^2 ) = 666.6666 nanoseconds. For the moving train the interval is equal to the time as X' is always equal to zero as the train engineer is always in the cab in front of the train.

 

[Orodruin]

Why is it that in SR we always seem to jump to the Lorenz equations when there is a simpler way.

We don't. The problem you refer to would be solved most easily by the time-dilation formula, which is of course equivalent to both methods. For a constant velocity $v$ you would obtain:
$$
(t')^2 = t^2 - x^2 = t^2 (1 - v^2) \quad \Longrightarrow t' = \frac{t}{\gamma}.
$$
Naturally, it does not matter what approach you use, the end result is consistent.

 

[robphy]

It's fair to say that different people attack problems based on the context given (are you given components or intervals?) and their own comfort-level.

Given a geometry problem [including, say, free-body diagrams and problems in special-relativity],
some will work with...
coordinates [possibly transformed by rotations],
components via right-triangle trigonometry,
magnitudes-and-angles with the laws of sin and cosine and tangent,
vectors with dot- and cross-products,
tensors, differential forms, etc...

I think it's also fair to say that the typical physicist encounters relativity
closer to the spirit of the physicist Einstein rather than the mathematician Minkowski.
That is, they think more in terms of moving frames of reference [train cars in relative motion in space]
rather than in terms of the ([gasp] non-Euclidean) spacetime geometry on a spacetime diagram.
Just look at the typical physics textbook.

So, I sympathize somewhat with the OP.
However, i think there are many ways that one can attack a problem [and it's good to get practice doing so in as many ways as possible].
In addition, one should keep your audience in mind.

 

[Orodruin]

That is, they think more in terms of moving frames of reference [train cars in relative motion in space]
rather than in terms of the ([gasp] non-Euclidean) spacetime geometry on a spacetime diagram.
Just look at the typical physics textbook.

I strongly disagree. As a theoretical physicist I am very much in favour of a general geometric description, which is often way more elegant and conveys the structure of the theory in a better way. That the typical undergraduate textbook will present things in terms of moving reference frames is a completely separate matter.

 

[robphy]

I strongly disagree. As a theoretical physicist I am very much in favour of a general geometric description, which is often way more elegant and conveys the structure of the theory in a better way. That the typical undergraduate textbook will present things in terms of moving reference frames is a completely separate matter.

I too am a theoretical physicist in favor of a general geometric description.
Unfortunately, the typical physicist is not a theoretical physicist...
and
there are likely many theoretical physicists that don't think about relativity like a relativist does.

 

[arydberg]

I too am a theoretical physicist in favor of a general geometric description.
Unfortunately, the typical physicist is not a theoretical physicist...
and
there are likely many theoretical physicists that don't think about relativity like a relativist does.

 

[arydberg]

I posted this because to me simpler is better and the interval seem the simplest way of trying to understand SR. I also tend to think there is more to study in SR with one subject being the magnetic field is a result of SR.

 

[Dale]

it's good to get practice doing so in as many ways as possible]

I agree. They are just different tools for the toolbox. Everybody has their favorite, but they are all useful and you should be familiar with all of them even if you have a preference.

 

[robphy]

I posted this because to me simpler is better and the interval seem the simplest way of trying to understand SR. I also tend to think there is more to study in SR with one subject being the magnetic field is a result of SR.

In the spirit of emphasizing
intervals over observer-dependent time- and space-components and their transformations,
one should also emphasize
the electromagnetic field tensor over the magnetic field, which is a set of observer-dependent components of that tensor.

 

[vanhees71]

I'd say, using the abstract geometric point of view (where geometry is to be understood as a very wide concept in the sense of Klein's program) of theoretical/mathematical physics to formulate fundamental laws. E.g., in my opinion the fundamental laws of electromagnetism are much easier to grasp using the manifest covariant four-vector formalism (or even more modern the Cartan calculus of differential forms).

However, when it comes to "bread-and-butter physics", describing concrete physical situations, you need the intuitive pictures to describe the given situation in a definite reference frame in the (1+3) formalism. E.g., take the (in principle very simple) problem to calculate the electromagnetic field of a uniformly moving point charge (in special relativity). You can do that (perhaps, I've not tried yet) in the manifestly covariant formalism (in Lorenz gauge!), but it's much simpler to calculate the field in the rest frame of the particle (just having a simple electrostatic Coulomb field) and then Lorentz boost the solution.