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Lorentz Invariants

In the previous handout we encountered the notion of 4-vectors, the prototype of which is the space-time vector, , where the ``zeroth component'' is time multiplied by the speed of light: and the remaining three components are the three ordinary spatial coordinates. [The notation is new but the idea is the same.] In general we assume that a vector with Greek indices (like ) is a 4-vector, while a vector with Roman indices (like ) is an ordinary spatial 3-vector.

It can be shown gif that the inner or scalar product of any two 4-vectors has the agreeable property of being a Lorentz invariant - i.e. it is unchanged by a Lorentz transformation - i.e. it has the same value for all observers. This comes in very handy in the confusing world of Relativity! We write the scalar product of two 4-vectors as follows:

where the first equivalence expresses the Einstein summation convention - we automatically sum over repeated indices. Note the - sign! It is part of the definition of the ``metric'' of space and time, just like the Pythagorean theorem defines the ``metric'' of flat 3-space in Euclidean geometry.

Our first Lorentz invariant was the proper time of an event, which is just the square root of the scalar product of the space-time 4-vector with itself:

We now encounter our second 4-vector, and probably the last one we need for our purposes here; the energy-momentum 4-vector,

where is the total relativistic energy and is the usual momentum 3-vector of some object in whose kinematics we are interested. [Check for yourself that all the components of this vector have the same units, as required.] If we take the scalar product of with itself, we get a new Lorentz invariant:

where is the square of the magnitude of the ordinary 3-vector momentum.

It turns outgif that the constant value of this particular Lorentz invariant is just the times the square of the rest mass of the object whose momentum we are scrutinizing: or . As a result, we can write

 

which is a very useful formula relating the energy E, the rest mass m and the momentum p of a relativistic body.

Although there are lots of other Lorentz invariants we can define by taking the scalar products of 4-vectors, these two will suffice for my purposes; you may forget this derivation entirely if you so choose, but I will need Eq. (E=mc2.14) for future reference.





next up previous
Next: The Mass of Up: No Title Previous: Conversion of Energy



Jess Brewer
Fri Sep 13 11:17:01 PDT 1996