This document summarizes key principles of Einstein's theory of special relativity, including: 1) Special relativity is based on two postulates - the principle of relativity and the constant speed of light. This leads to time dilation, length contraction, and the equivalence of mass and energy. 2) Lorentz transformations relate the coordinates of observers in different inertial reference frames and explain how the speed of light remains constant for all observers. 3) Spacetime diagrams illustrate how events that are simultaneous for one observer may not be for another due to the finite speed of light and relativity of simultaneity.

1. General Relativity
Bernard Schutz

2. Principles of Special Relativity
• Based on Lorentz Transformation
Time Dilation/Lorentz Contraction
Mass to Energy
• Special Relativity
Deduced from
Principle of Relativity (Galileo)
We can only measure relatives
Speed of Light
Speed of light always 3*10^8 to an unaccelerated
observer
• Inertial Observer
Constant velocity with respect to any other unaccelerated observer

3. Principles of Special Relativity
• Galilean Law of Addition of Velocities
v(t)v’(t)=v(t)-V
If I’m measuring speed of runner at 5 m/s, and I’m
moving 1 m/s, speed of runner is 4 m/s relative to me
Einstein finding light as a constant velocity crushes this
If light moves at 3*10^8, and I’m moving at
2*10^8, then why does light still appear to be moving
3*10^8
• Einstein’s Special Relativity vs. General Relativity
SR only works when discounting gravitational fields
GR extends SR to account for gravity

4. Lorentz Transformation
• Mathematics that explains how the speed of light
is independent of reference frame
• For relativity, describes relationship between two
observers
• Assuming two observers in standard
configuration, a Lorentz Transformation in Special
Relativity will relate (t,x,y,z) to (t’,x’,y’,z’)
Symmetry of a Lorentz Transformation makes the
inverse easy to find in Special Relativity, and
explains that all physical laws are constant under a
Lorentz Transform

5. Inertial Observer
• An observer is an information gathering system
Inertial coordinate is best described as a set point (t, x, y, z)
Must satisfy three principles
P2-P1 must be independent of time
Clocks are synchronized, run at same rate
Geometry at any given time, t, is Euclidian
*Must be unaccelerated, because only unaccelerated
observers can keep clocks synchronized*
• Observation
Finding x, y, z, t or an event
t isn’t observed time, but rather time event happened (usually)
• Inertial reference frame=inertial observer

6. Units
• In Special Relativity, we often take c to have a
value of 1 unit of time
Lights years, meters (of time), light seconds,
etc…

7. Spacetime Diagram
• Imagine an x-t diagram (single point is an event)
• World line is x=x(t)
Slope=dt/dx=1/v
• Light ray has a 45 degree angle and a slope of 1
• Notes
Observer is O, events are Greek letters (sometimes)
Coordinates are (t,x,y,z), all in meters
We can define coordinates as (x^0,x^1,x^2,x^3), so for
example (x^2) is also the y coordinate
Latin indexes are used for spatial coordinates, be sure to
check

8. Coordinates of Multiple Observers
• Since observers exist in the same spacetime, we should
be able to indicate observations of separate observers
on a single spacetime diagram
• Imagine two observers, O=(x,t), and O’=(x’,t’)
t’ axis
t’ is the beginning of events for O’, when x’=0; for
simplicity, t=t’=0 (both events start at same time)
x’ axis
The x’ position at t’=0 (for simplicity x’=t’=0=starts
at the origin)

9. Coordinates of Multiple Observers
• Light ray for O’ on time diagram for O’
Imagine light, starting at one position, traveling a
certain distance, reflecting, then returning to the
original position
Time to return after reflecting will be the
same as time to reach the reflecting medium, or t=-
a and t=a ( l-al=lal )
By defining t=a and t=-a, we can use the
x-axis to define origin of reflection of light ray

10. Coordinates of Multiple Observers
• Light ray for O’ on O
Using t’ axis from before, we arbitrarily place event ε at t=-a, and event
Ρ at t=a
We will then emit a light ray at a 45 degree angle (due to slope
of 1) from event ε
We will then draw 45 degree angle (slope of -1) from
event P to light ray to form event θ
We then draw a 45 degree angle (slope of 1)
through point θ, which will be the x’ axis
We find by these methods that events simultaneous to O are not
simultaneous to O’
Called a failure of simultaneity, unavoidable
From the perspective of O, O’ will move to the right, while from the
perspective of O’, O will move to the left

11. Invariance of the Interval
• We have found position, but not the length scale
A particular length or distance
Physical phenomena of differing length scales
decouple
• We find interval of events to be
Since we defined the slope of light to be equal to
one, then this equation equals 0
Ds2
= -(Dt)2
+(Dx)2
+(Dy)2
+(Dz)2
Ds2
= -(Dt)2
+(Dx)2
+(Dy)2
+(Dz)2

12. Invariance of the Interval
• Finding the relationship between two frames
First we must assume O and O’ are linear and
originate from the same point, giving
Since both frames are linear, we find t’,x’,y’,z’
to be linear combinations of t,x,y,z, so the equation
is a quadratic function
Therefore (??????)
j
Ds-2
= -(Dt')2
+(Dx')2
+(Dy')2
+(Dz')2
s-2
= Mab
b=0
3
å (Dxa
)(Dxb
)
a=0
3
å

13. Invariance of the Interval
• Since delta S^2=0 due to the slope of light being 1, we can
solve for delta t and assign
• We can then use the quadratic form for when delta s^2 is 0
(light is unbended)
• Since delta s^2 will vanish, any arbitrary x must vanish as
well, so we find
• δ is the Kronecker delta, defined as a 1 when i=j and a 0
when i=/=j
Dt = Dr = Dx2
+Dy2
+Dz2
Ds-2
= M00 (Dr2
)+ 2( M0iDxi
)Dr + MijDxi
Dx j
j=1
3
å
i=1
3
å
i=1
3
å
M0i = 0,i =1,2,3
Mij = -(M00 )d,(i, j =1,2,3)

14. Invariance of the Interval
• Ultimately, we find
• We can solve replace Moo by a phase shift with
respect to v, so
• And
• Since Δs^-2=Δs^2, we can ultimately conclude
Φ(v) is equal to one, but I’m sure there’s a more
complex solution we will use…….to be continued
Ds-2
= M00(Dt2
-Dx2
-Dy2
-Dz2
)
M00 =f(v)
Ds-2
=f(v)Ds2

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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
)

21. 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
)

22. 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

23. 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

24. 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