Bernard-Schutz General Relativity

1. Invariance of the Interval
• φ depends only on speed
Imagine an object with length, l, perpendicular to the x-
axis
In spacetime diagram, draw in ends and shade area
between to represent length
Clock is parallel to t-x diagram of O’ on O
spacetime diagram, perpendicular to y axis
Both ends are simultaneous since
there’s no change in x, z, or t axis for observer O
Choose clock that passes
midway point of Α and β

2. Invariance of the Interval
• A light ray will reflect back to some exact point
after some time, meaning Α and β are
simultaneous for O’
Therefore, Length of rod O’^2 is related to
length of rod of O^2 by φ(v)
Since rod is perpendicular, there is no
preferred direction and φ(v) is a scalar quantity

3. Invariance of the Interval
• Imagine O, O’, and O’’ which moves opposite of O’
We find s’’^2=s’^2=s^2, so φ(v) is plus or minus one
Since we’re using squares, we take the positive
value
• Length perpendicular to relative velocity is the same to
all observers
Any event in a frame perpendicular to motion is
simultaneous for viewers
If I’m moving to left and friend is moving to right,
an object in our midpoint moving vertically will be
simultaneous

4. Invariance of the Interval
• Δs relates events, not observers
If positive (Latin>Greek) events are spacelike separated (Me and an
alien doing a jumping jack 3,000,000 light years away)
If negative (Greek>Latin) events are timelike separated (Me and an
alien doing a jumping jack 3,000,000 nanometers away)
If 0, events are lightlike/null separated
• Light Cone of A
Events inside are timelike separated, outside are spacelike separated,
lines are null separated
Quadrants represent absolute future (+Δt), absolute past (-Δt) and
elsewhere (outside of light cone)
Events inside the light cone are reachable by physical object
Past/future of certain objects can overlap but will not be the
same

5. Invariant Hyperbolae
• Way of calibrating x’ and t’ in O reference frame
• Consider constant motion a=-t^2+x^2 for x-t diagram
Due to invariance of interval, we find Δa’=Δa
• Hyperbolae are drawn with a slope approaching that of light
Since a=-1=-t^2+x^2, on t axis (where x=0), it follows that t=1
Since a’=a, t=t’=1 we can find event β at t’=1
Same logic to find x’ axis
• Once again, interval is more important than anything (Δs)
• Revelations of SR
Adds time coordinate in distance calculations
In our everyday life, events seem simultaneous
• Line of Simultaneity
Line where events will be simultaneous (line is tangent to event),
Slope of line is velocity of frame

6. Results
• Time Dilation
As we see, t=1 and t’=1 are defined at different points
t’ seems slower since it is further vertically from
the origin
Proper Time
Time measured that passes through both events
We find –Δτ^2=-Δt’^2 when clock is moving at
same speed as O’ (clock is at rest), and by finding in terms
of coordinates we get:
dDt =
Dt
(1-v2
)

7. Results
• Lorentz Contraction
Imagine a rod at rest along O’ inertial frame
Length for O is Δs^2 along x axis, Δs’^2 along x’ axis
From calculation we find
As we approach the speed of light, an object will contract
• Interval Δs
No universal agreement on definition (positive or
negative); however this is irrelevant due to invariance
Make sure to check what is being solved
XB =1 (1-v2
)

8. Results
• Failure of Simultaneity can often lead students
to believe that finite transmission signal can
cause time dilation
This is due to two people defining “now” as a
concrete time, but not agreeing on what “now”
is, a consequence of the speed of light being a
finite limit
Always important to keep in mind that
time is a coordinate, not universal

9. Lorentz Transformation
• Assuming y’=y and z’=z, we find
t’=αt+βx
x’=γt+δx
With α, β, γ, δ all dependent on velocity
• Due to axis equations (t’=vt-x, x’=vx-t, x’=t’=0), we can infer that
γ/δ and β/α are -1
Because of this, we can express t’=α(t-xv)
We can take invariance of Δs to give
α= , so we take positive sign
This gives complete transform as t’=αt-αvx, with using value of α as
previously given value
This is called a boost of velocity in x
This transformation only works without needing rotation
(1 (1-v2
))2

10. Velocity-Composition Law
• Example of using Lorentz Transformation to
derive rules of SR
• We find speed never exceeds light if v<c
• We also find small velocities can be accurately
predicted using Classical Mechanics
This justifies Galilean Law of Addition of
Velocities at v<<c