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Consider a matrix transformation $\mathbf{T}$ that acts on a vector $\mathbf{x}$:

$$ \mathbf{x}' = \mathbf{T}\mathbf{x}. $$

Now, I know that one-dimensional linear transformations expand the length by a factor $|det(\mathbf{T})|$, two-dimensional linear transformations expand the area by a factor $|det(\mathbf{T})|$, and three-dimensional linear transformations expand the volume by a factor $|det(\mathbf{T})|$.

How can I see this mathematically though? I was thinking of calculating the norm $||\mathbf{x}'|| = ||\mathbf{T}||\cdot||\mathbf{x}||$ but don't know how to deal with $||\mathbf{T}||$

The physics of this question is: do all Lorentz transformations have determinant equal to 1? Because they preserve the space-time interval?

EDIT:

What I was asking was more about: how can I see that the determinant of a matrix carries information about the area/volume change of the system it is acting on? Mathematically, how can I show that the area spanned by two vectors is unchanged IFF the determinant of the transformation matrix is 1?

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  • $\begingroup$ Thanks. But still, what is $||\mathbf{T}||$? $\endgroup$
    – SuperCiocia
    Aug 31, 2014 at 20:17
  • $\begingroup$ If $\mathbf{T} = \begin{array}{cc} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{array} $, what is $\mathbf{T}$? $\endgroup$
    – SuperCiocia
    Aug 31, 2014 at 20:23
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    $\begingroup$ How can I see this mathematically though?...uh, isn't that what your first equation does: $\mathbf x'=\mathtt T\mathbf x$? $\endgroup$
    – Kyle Kanos
    Aug 31, 2014 at 20:57
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    $\begingroup$ The linear algebra/calculus book by Hubbard and Hubbard illustrates this extremely well, for whatever that's worth. $\endgroup$
    – DanielSank
    Sep 8, 2014 at 23:02
  • $\begingroup$ There is a book by Felix Klein you might find interesting. "Elementary Mathematics from an Advanced Standpoint: Geometry". There is also one on arithmetic and analysis (I think a combination of two volumes). These were written for classes to teach high school teachers what kind of math was needed to prep for science and math in the colleges. From 1939 and sounds like a very familiar complaint. The exploration of transformations and their geometry is pretty cool. I'm old enough that my physics profs were steeped in projective geometry and taught a bit of it since the math departments did not. $\endgroup$ Sep 8, 2014 at 23:10

2 Answers 2

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do all Lorentz transformations have determinant equal to 1? Because they preserve the space-time interval?

Yes, they do, but preservation of the interval is not the right way to think about the "why."

We have two facts:

  1. Lorentz transformations have Jacobian determinant 1.

  2. Lorentz transformations preserve the spacetime interval.

There is no close or simple relationship between these facts. In particular, 1 does not imply 2. For example, the Galilean transformations have Jacobian determinant 1, but they do not preserve the spacetime interval. Similarly, we could define a rotation in the $x-t$ plane, and it would have Jacobian determinant 1 but not preserve the spacetime interval.

It is true that 2 implies 1, but this is simply because 2 is sufficient to completely characterize the Lorentz transformations and therefore give all their properties indirectly.

For a general proof of the unit Jacobian that applies to both Galilean transformations and Lorentz transformations, see the answer to this question: Motivation for preservation of spacetime volume by Lorentz transformation?

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Yes, the determinant of the matrix expressing the components of a Lorentz transformation is dictated by the need to keep the spacetime interval invariant.

In the following I will use natural units such that $c=1$, use the "East coast" sign convention, and use the Einstein summation convention. Primed indices indicate vector or tensor components as expressed in the primed coordinate system.

In any coordinate system considered by special relativity, the spacetime interval between two events can be written

$$s^2=\ -\ (\Delta t)^2\ +\ (\Delta x)^2\ +\ (\Delta y)^2\ +\ (\Delta z)^2\ \ .$$

This means that if we define

$$A=\left\lbrack \begin{matrix}\Delta t \\ \Delta x \\ \Delta y \\ \Delta z \end{matrix}\right\rbrack$$

we can write the spacetime interval as

$$s^2=A^{\mu} g_{\mu\nu} A^{\nu}\ \ ,$$

where $g$ is the Minkowski metric, whose components are:

$$g_{\mu\nu}=\left\lbrack \begin{matrix}-1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{matrix}\right\rbrack \ \ .$$

The Minkowski metric is what defines the geometry of the spacetime used in special relativity, Minkowski space. The Minkowski metric defines what intervals are, in the same way that the Euclidean metric defines what lengths are in Euclidean space. A key point about the Minkowski metric is that its components are the same in any coordinate system considered by special relativity. The invariance of intervals is a direct, easy consequence of the invariance of the components of the Minkowski metric.

To convert a vector from one coordinate system to another, you use a Lorentz transformation once:

$$x^{\mu '}={\Lambda^{\mu '}}_{\nu}x^{\nu}\ \ .$$

But the Minkowski metric is a rank two tensor. To convert a rank two tensor from one coordinate system to another, you use a Lorentz transformation twice, like

$$g_{\mu' \nu'}={\Lambda^{\sigma}}_{\mu '}g_{\sigma\rho}{\Lambda^{\rho}}_{\nu '}\ \ ,$$

which you could write in matrix notation as

$$g'\ =\ {\Lambda}^{T} g \Lambda\ \ .$$

But since the components of the Minkowski metric are the same in any coordinate system, we can also write

$$g\ =\ {\Lambda}^{T} g \Lambda\ \ .$$

Taking the determinant of both sides of that equation gives

$$\det(g)=\det(\Lambda)\det(g)\det(\Lambda)\ \ .$$

The only way for that equation to hold is if

$$[\det(\Lambda)]^2=1\ \ .$$

There are two possible ways that that equation can hold. The most commonly considered possibility is that

$$\det(\Lambda)\ =\ 1\ \ ,$$

in which case the transformation is called a "proper" Lorentz transformation. The other possibility is that

$$\det(\Lambda)\ =\ -1\ \ ,$$

in which case the transformation is called an "improper" Lorentz transformation. Improper Lorentz transformations are less frequently useful than proper Lorentz transformations.

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