Chapter 1 - Special relativity and spacetime 3
Basic concepts of special relativity 3
Event 3
Frame of reference 3
Observe 3
Special relativity 3
Coordinate Transformation 4
Standard Configuration 4
Galilean Transformation 4
Lorentz Transformation 5
Inverse Lorentz Transformation 5
Interval Transformation 6
Inverse Interval Transformation 6
Consequences of the Lorentz transformations 6
Time Dilation 6
Length Contraction 7
Relativity of Simultaneity 7
Doppler Effect 7
Velocity Transformation 8
Minkowski Spacetime 8
Spacetime diagrams 8
Spacetime Separation & Minkowski Metrics 8
Separation Relationships 9
Chapter 2 - Special relativity and physical laws 10
Invariants and physical laws 10
Invariance of Physical Quantities 10
Invariance of Physical Laws 10
The laws of mechanics 10
Relativistic Momentum 10
Relativistic Kinetic Energy 10
Total Relativistic Energy & Mass Energy 11
Four-Momentum 11
Four-component 11
Four-velocity 11
Four-momentum 11
Four-force 12
Four-Vectors 12
Energy-Momentum Relation 14
Laws of electromagnetism 14
Conservation of charge 14
Four-Current 14
Four-Tensor 15
1
,Chapter 3 - Geometry and Curved Spacetime 15
Line Elements and Differential Geometry (Euclidean) 15
Line Element in a Plane 15
Curved Surfaces 16
Metrics and connections 17
Metrics and Riemannian geometry 17
Connections and Parallel Transport 17
Geodesics 17
Most Direct Path 18
Shortest Distance 18
Curvature 18
Curvature of a Curve in a Plane 19
Gaussian Curvature of a 2-Dimensional Surface 19
Curvature in Spaces of Higher Dimensions 20
Chapter 4 - General Relativity and Gravitation 20
3 Founding Principles of General Relativity 20
Equivalence 20
Weak Equivalence Principle 20
Strong Equivalence Principle 20
General Covariance 20
Rules of Tensor Algebra 21
Scaling 21
Addition & Subtraction 22
Multiplication 22
Contraction 22
Contravariant Differentiation 22
Covariant Differentiation 22
Consistency 22
Ingredients of General Relativity 23
Energy-Momentum Tensor 23
Ideal Fluid 23
Einstein Tensor 24
Einstein’s Field Equation & Geodesic Motion 24
Field Equation 24
Geodesic Motion 24
2
, APM3713 Summary
Chapter 1 - Special relativity and spacetime
Basic concepts of special relativity
Event
Instantaneous occurance at a specific point in space
➔ Coordinates: t (time) and x, y, z (crtisian) or r, 𝛳, Φ (spherical)
➔ Time of event is time where event happens, NOT time when it is seen from some far off
point
Frame of reference
System for assigning coordinates to an event
➔ Represented by letter S
➔ Inertial frames: a frame of reference in which a body that in not subject to any net force
maintains a constant velocity (aka Newton's first law of motion holds true)
➔ Frames that move with constant velocity relative to the inertial frame are also inertial
frames
(can only be inertial frame if it doesn't accelerate or rotate relative to the inertial frame)
Observe
An individual dedicated to using a particular frame of reference for recording events
➔ Represented by letter O
➔ Location of observer is not important for reporting an events coordinates
➔ Inertial Observer: uses inertial frame of reference to observe events
Special relativity
➔ Postulates (statement believed to hold true):
◆ Principle of relativity: the laws of physics can be written in the same form in all
inertial frames
◆ Principle of the Constancy of the Speed of Light: Speed of light in a vacuum
has the same constant value in all inertial frames
8 −1
(𝑐 = 3 × 10 𝑚. 𝑠 ). Meaning light (or electromagnetic signals that travel at the
speed of light) can be used to ensure that all clocks are properly synchronized
3
Basic concepts of special relativity 3
Event 3
Frame of reference 3
Observe 3
Special relativity 3
Coordinate Transformation 4
Standard Configuration 4
Galilean Transformation 4
Lorentz Transformation 5
Inverse Lorentz Transformation 5
Interval Transformation 6
Inverse Interval Transformation 6
Consequences of the Lorentz transformations 6
Time Dilation 6
Length Contraction 7
Relativity of Simultaneity 7
Doppler Effect 7
Velocity Transformation 8
Minkowski Spacetime 8
Spacetime diagrams 8
Spacetime Separation & Minkowski Metrics 8
Separation Relationships 9
Chapter 2 - Special relativity and physical laws 10
Invariants and physical laws 10
Invariance of Physical Quantities 10
Invariance of Physical Laws 10
The laws of mechanics 10
Relativistic Momentum 10
Relativistic Kinetic Energy 10
Total Relativistic Energy & Mass Energy 11
Four-Momentum 11
Four-component 11
Four-velocity 11
Four-momentum 11
Four-force 12
Four-Vectors 12
Energy-Momentum Relation 14
Laws of electromagnetism 14
Conservation of charge 14
Four-Current 14
Four-Tensor 15
1
,Chapter 3 - Geometry and Curved Spacetime 15
Line Elements and Differential Geometry (Euclidean) 15
Line Element in a Plane 15
Curved Surfaces 16
Metrics and connections 17
Metrics and Riemannian geometry 17
Connections and Parallel Transport 17
Geodesics 17
Most Direct Path 18
Shortest Distance 18
Curvature 18
Curvature of a Curve in a Plane 19
Gaussian Curvature of a 2-Dimensional Surface 19
Curvature in Spaces of Higher Dimensions 20
Chapter 4 - General Relativity and Gravitation 20
3 Founding Principles of General Relativity 20
Equivalence 20
Weak Equivalence Principle 20
Strong Equivalence Principle 20
General Covariance 20
Rules of Tensor Algebra 21
Scaling 21
Addition & Subtraction 22
Multiplication 22
Contraction 22
Contravariant Differentiation 22
Covariant Differentiation 22
Consistency 22
Ingredients of General Relativity 23
Energy-Momentum Tensor 23
Ideal Fluid 23
Einstein Tensor 24
Einstein’s Field Equation & Geodesic Motion 24
Field Equation 24
Geodesic Motion 24
2
, APM3713 Summary
Chapter 1 - Special relativity and spacetime
Basic concepts of special relativity
Event
Instantaneous occurance at a specific point in space
➔ Coordinates: t (time) and x, y, z (crtisian) or r, 𝛳, Φ (spherical)
➔ Time of event is time where event happens, NOT time when it is seen from some far off
point
Frame of reference
System for assigning coordinates to an event
➔ Represented by letter S
➔ Inertial frames: a frame of reference in which a body that in not subject to any net force
maintains a constant velocity (aka Newton's first law of motion holds true)
➔ Frames that move with constant velocity relative to the inertial frame are also inertial
frames
(can only be inertial frame if it doesn't accelerate or rotate relative to the inertial frame)
Observe
An individual dedicated to using a particular frame of reference for recording events
➔ Represented by letter O
➔ Location of observer is not important for reporting an events coordinates
➔ Inertial Observer: uses inertial frame of reference to observe events
Special relativity
➔ Postulates (statement believed to hold true):
◆ Principle of relativity: the laws of physics can be written in the same form in all
inertial frames
◆ Principle of the Constancy of the Speed of Light: Speed of light in a vacuum
has the same constant value in all inertial frames
8 −1
(𝑐 = 3 × 10 𝑚. 𝑠 ). Meaning light (or electromagnetic signals that travel at the
speed of light) can be used to ensure that all clocks are properly synchronized
3
