Show that gμν g^μν=4. Examples of scalar products are p^μxμ≡p ·x=E t-𝐩 ·𝐱 p^μpμ≡p ·p

Show that gμν g^μν=4. Examples of scalar products are ...

Show that $g_{\mu \nu} g^{\mu \nu}=4$. Examples of scalar products are $$ \begin{aligned} & p^\mu x_\mu \equiv p \cdot x=E t-\mathbf{p} \cdot \mathbf{x} \\ & p^\mu p_\mu \equiv p \cdot p \equiv p^2=E^2-\mathbf{p}^2 . \end{aligned} $$ These quantities are Lorentz invariants. For a free particle, we have $p^2=m^2$, see (3.10). We say that the particle is on its mass shell.

Show that $g_{\mu \nu} g^{\mu \nu}=4$.
Examples of scalar products are
$$
\begin{aligned}
& p^\mu x_\mu \equiv p \cdot x=E t-\mathbf{p} \cdot \mathbf{x} \\
& p^\mu p_\mu \equiv p \cdot p \equiv p^2=E^2-\mathbf{p}^2 .
\end{aligned}
$$

These quantities are Lorentz invariants. For a free particle, we have $p^2=m^2$, see (3.10). We say that the particle is on its mass shell.

Key Concepts

Lorentz Invariance

Lorentz invariance is a foundational principle of relativity stating that the laws of physics are the same in all inertial frames. This invariance is evident in the preservation of scalar products, such as p · x and p², under Lorentz transformations, ensuring that physical quantities like the mass of a free particle remain constant.

Minkowski Spacetime

Minkowski spacetime is the arena for special relativity, characterized by a flat four-dimensional space with a metric signature that distinguishes time and space components. This framework underpins the concepts of Lorentz invariance and the invariant spacetime intervals between events.

Metric Tensor

The metric tensor is the mathematical object that encodes the geometric properties of a space or spacetime, including how distances and angles are measured. In special relativity, the Minkowski metric tensor defines the spacetime structure, allowing for the calculation of intervals and the lowering and raising of indices in tensor expressions.

Tensor Contraction

Tensor contraction is the process of summing over one upper and one lower index of a tensor, which reduces its rank and can yield a scalar quantity. When the metric tensor is contracted with its inverse, as in g???? g????, the result is a trace that reflects the number of dimensions in the space.

Transcript

00:03 In this problem, we want to prove that e squared minus p squared c squared doesn't change or is invariant under a lorentz transformation.

00:26 So to prove this, we'll start out by writing what the lorentz transformation for energy and momentum is.

00:33 So we can conveniently write this as e and pc equals gamma minus gamma beta, minus gamma beta again, gamma, and that's signs epc, or excuse me, these are the modified, lorenz transformed ones, so e prime, p prime.

01:06 So what we're trying to prove here then is that e squared minus p squared c squared is equal to our lorentz transformed versions of these things.

01:20 So e prime squared minus p prime squared squared, c squared.

01:30 And for this, we're of course relying on the definitions that beta is our velocity over the speed of light.

01:37 And gamma is 1 divided by square root of 1 minus our velocity over the speed of light and that all that's squared.

01:46 So beta square.

01:48 So we'll start by and multiplying out these matrices to get an expression for e prime and p prime c.

01:55 So if we do that, we'll get that e prime is gamma e.

02:01 So gamma e minus gamma beta p c gamma beta pc and similarly for p p p c this is just going to be e minus gamma b or minus gamma b beta excuse me beta not p minus gamma beta plus gamma p c and now that we have these expressions we can square them and plug them into our equation here.

02:41 So if we do that and we'll omit this side of the equation for now just to make it more simple to look at.

02:49 So e prime squared is going to be gamma e minus gamma beta p c and that entire quantity squared is going to be minus p prime c squared.

03:06 So that's going to be minus gamma beta e plus gamma p c.

03:16 The whole thing's squared.

03:18 And we'll foil all of this out to get gamma squared, e squared, minus 2e.

03:30 And notice we have a gamma on both parts of this equation.

03:34 So we'll have a gamma squared there.

03:37 And then the rest of that part.

03:39 So beta p c and that's going to be plus gamma squared, beta squared, p squared, c squared for this section...