Intervals
LL2 section 2
4-space is the set of all world points
described by 3 space coordinates and time
• Every particle has a “world line”.
• Uniform rectilinear motion has a straight
t
world line.
x
Consider two events that are described with
respect to two inertial reference systems
As described in system K
Event #1: light pulse emitted from (x1, y1, z1) at t1
Event #2: light pulse arrives at (x2,y2,z2) at t2
Z’
Z
V
t
K
Y
K’
X
Y’
t’
X’
Proof
The interval between two events is
•1 The same in all inertial reference frames
•2 Is different in every inertial reference frame
•3 Might be the same or different, depending on
which inertial frames are considered.
The interval between two events is
• The same in all inertial reference frames
• Is different in every inertial reference frame
• Might be the same or different, depending on
which inertial frames are considered.
Distance between points where
events occur at the same time:
The condition for two events to occur at the
same point in some reference frame is that
• The interval between them is real.
2• The interval between them is imaginary.
3• The interval between them is zero.
1
The condition for two events to occur at the
same point in some reference frame is that
• The interval between them is real.
• The interval between them is imaginary.
• The interval between them is zero.
The condition for two events to occur at the
same time in some reference frame is that
•1 The interval between them is real.
•2 The interval between them is imaginary.
•3 The interval between them is zero.
The condition for two events to occur at the
same time in some reference frame is that
• The interval between them is real.
• The interval between them is imaginary.
• The interval between them is zero.
• Impossible to find any frame where two
events separated by a time-like interval are
simultaneous.
• Can’t find any frame where even occurs with
t<0.
• Can’t make P occur simultaneously or before
O
• Opposite lower region defines absolute past
relative to O.
Events can be related causally if
interval between them is
•1 Zero
•2 Space like
•3 Time like
Events can be related causally if
interval between them is
• Zero
• Space like
• Time like