Tensors and Relativity: Chapter 2

Four- vectors

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A four- vector is a quantity with four components which changes like spacetime coordinates under a coordinate transformation. We will write the displacement four- vector as:

equation678

tex2html_wrap_inline1202 is a frame independent vector joining near by points in spacetime. tex2html_wrap_inline1204 means: has components in frame tex2html_wrap_inline1206 . tex2html_wrap_inline1208 are the coordinates themselves [ which are coordinate dependent ].

In frame tex2html_wrap_inline1210 , the coordinates are tex2html_wrap_inline1212 so:

equation689

The Lorentz transformations  can be written as

equation696

where tex2html_wrap_inline1214 is the tex2html_wrap_inline1216 Lorentz transformation matrix

eqnarray701

tex2html_wrap_inline1218 and tex2html_wrap_inline1220 , tex2html_wrap_inline1222 can be regarded as column vectors.

The positioning of the indices is explained later but indicates that we can use the  summation convention: sum over repeated indices, if one index  is up and one index down. Thus we can write:

equation705

tex2html_wrap_inline1224 is a dummy- index  which can be replaced by any other index; tex2html_wrap_inline1226 is a free- index , so the above equation is equivalent to four equations. For a general four- vector

equation710

we can write

equation715

If tex2html_wrap_inline1228 and tex2html_wrap_inline1230 are two four- vectors, clearly tex2html_wrap_inline1232 and tex2html_wrap_inline1234 are also four- vectors with obvious components

equation731

and

equation736

In any frame tex2html_wrap_inline1206 we can define a set of four- basis vectors :

equation742

equation745

equation748

and

equation751

In general we can write

eqnarray754

where tex2html_wrap_inline1238 labels the basis vector and tex2html_wrap_inline1224 labels the coordinate.

Any four- vector can be expressed as a sum of four- vectors parallel to the basis vectors i.e.

eqnarray756

The last equality reflects the fact that four- vectors are frame independent.

Writing:

equation762

where we have exchanged the dummy indices tex2html_wrap_inline1238 and tex2html_wrap_inline1224 and tex2html_wrap_inline1226 and tex2html_wrap_inline1248 , we see that this equals tex2html_wrap_inline1250 for all tex2html_wrap_inline1228 if and only if

equation776

This gives the transformation law for basis vectors :

eqnarray780

Note that the basis transformation law is different from the transformation law for the components since tex2html_wrap_inline1254 takes one from frame tex2html_wrap_inline1210 to tex2html_wrap_inline1206 .

So in summary, for vector basis  and vector components  we have:

eqnarray791

Since tex2html_wrap_inline1206 has a velocity tex2html_wrap_inline1262 relative to tex2html_wrap_inline1210 , we have:

eqnarray801

So tex2html_wrap_inline1266 is the inverse  of tex2html_wrap_inline1268 . Likewise

equation829

It follows that the Lorentz transformations with tex2html_wrap_inline1262 gives the components of a four- vector in tex2html_wrap_inline1206 from those in tex2html_wrap_inline1210 .

The magnitude  of a four- vector is defined as tex2html_wrap_inline1276 :

equation849

in analogy with the line element

equation851

The sign on the tex2html_wrap_inline1278 will be explained later. This is a frame invariant scalar.

tex2html_wrap_inline1228 is spacelike  if tex2html_wrap_inline1282 , timelike  if tex2html_wrap_inline1284 and null  if tex2html_wrap_inline1286 . The scalar product  of two four- vectors tex2html_wrap_inline1228 and tex2html_wrap_inline1230 is:

equation859

Since

equation865

tex2html_wrap_inline1292 is frame independent.

tex2html_wrap_inline1228 and tex2html_wrap_inline1230 are orthogonal if tex2html_wrap_inline1298 ; they are not necessarily perpendicular in the spacetime diagram [ for example a null vector is orthogonal to itself ], but must make equal angles with the tex2html_wrap_inline1300 line.

Basis vectors form an orthonormal tetrad  since they are orthogonal: tex2html_wrap_inline1302 if tex2html_wrap_inline1304 , normalized to unit magnitude: tex2html_wrap_inline1306 :

eqnarray887

We will see later what the geometrical significance of tex2html_wrap_inline1308 is.


next up previous index
Next: Four- velocity, momentum and Up: Title page Previous: Title page