28 Special Relativitytransformations between frames which are rotated or translated at a constant velocity
with respect to each other is then the line element of theMinkowski metric
d푠^2 =푐^2 d휏^2 =푐^2 d푡^2 −d푥^2 −d푦^2 −d푧^2 =푐^2 d푡^2 −d푙^2. (2.7)The trajectory of a body moving in space-time is called itsworld line. A body at a
fixed location in space follows a world line parallel to the time axis and, of course, in
the direction of increasing time. A body moving in space follows a world line making
a slope with respect to the time axis. Since the speed of a body or a signal travelling
from one event to another cannot exceed the speed of light, there is a maximum slope
to such world lines. All world lines arriving where we are, here and now, obey this
condition. Thus they form a cone in our past, and the envelope of the cone corresponds
to signals travelling with the speed of light. This is called thelight cone.
Two separate events in space-time can becausallyconnected provided their spatial
separation dland their temporal separation d푡(in any frame) obey|dl∕d푡|⩽푐.
Their world line is then inside the light cone. In Figure 2.1 we draw this four-
dimensional cone in 푡,푥,푦-space (another choice could have been to use the
coordinates푡,휎,휃). Thus if we locateour presentevent to the apex of the light cone
at푡=푡 0 =0, it can be influenced by world lines from all events inside thepastlight
cone for whichct<0, and it can influence all events inside thefuturelight cone for
whichct>0. Events inside the light cone are said to havetimelikeseparation from
the present event. Events outside the light cone are said to havespacelikeseparation
from the present event: they cannot be causally connected to it. Thus the light cone
encloses thepresent observable universe, which consists of all world lines that can in
principle be observed. From now on we usually mean the present observable universe
when we say simply ‘the Universe’.
For light signals the equality sign above applies so that the proper time interval in
Equation (2.7) vanishes
d휏= 0.
Events on the light cone are said to havenullorlightlikeseparation.
Redshift and Scale Factor. The light emitted by stars is caused by atomic transi-
tions with emission spectra containing sharp spectral lines. Similarly, hot radiation
traversing cooler matter in stellar atmospheres excites atoms at sharply defined wave-
lengths, producing characteristic dark absorption lines in the continuous regions of
the emission spectrum. The radiation that was emitted by stars and distant galaxies
with a wavelength휆rest=푐∕휈restat time푡in their rest frame will have its wavelength
stretched by the cosmological expansion to휆obswhen observed on Earth. Since the
Universe expands, this shift is in the red direction,휆obs>휆rest, and it is therefore called
aredshift, denoted
푧=
휆obs−휆rest
휆rest