Notes on Theory of Relativity
My raw notes - to be updated
This is my structured attempt to finally understand the theory of relativity — not just the scattered pieces I’ve picked up over time (like time dilation, Lorentz invariance, or something about tensors), but how it all fits together. What are the postulates? What are the components? What’s derived from what? This text is meant to give form to that mess and show the full skeleton.
Relativity is a physical theory of how space, time, matter, and energy behave — especially at high speeds or in gravitational fields. It’s not built like a formal mathematical theory with axioms and theorems, but it does start from a few physical postulates. These are backed by experiments, and they form the foundation for a consistent mathematical model of spacetime.
There are two parts:
Special Relativity (1905): Physics in flat spacetime (no gravity).
General Relativity (1915): Extension to curved spacetime (includes gravity).
What Was Known Before Einstein
The idea that "motion is relative" was not new. Galileo had already noted that the laws of motion work the same in any inertial frame - one that moves at constant velocity. (An inertial frame is like a “non-accelerating” viewpoint.)
Another example is, if you’re on a smoothly moving train and toss a balloon straight up, it behaves just as it would if the train were standing still.
This is the principle of relativity: the laws of physics are the same in all inertial frames. It was accepted long before Einstein.
What Einstein added was much more radical: he extended relativity to space and time themselves. In Newtonian physics, space and time were absolute — everyone agreed on durations, distances, and simultaneity. Einstein’s theory showed that:
Time and space are relative.
Observers in different frames will measure different durations, lengths, and even different sequences of events.
But the laws of physics — like Maxwell’s equations, or Newton’s second law — remain unchanged across inertial frames.
That’s the core idea of special relativity: keep the principle of relativity, but apply it to all of physics, even to the structure of space and time.
Later, with general relativity, Einstein went further: he asked what happens in non-inertial (accelerating) frames. That’s where the elevator thought experiment comes in:
Inside a sealed elevator, you can’t tell whether the force you feel is from gravity or from constant upward acceleration.
This insight — the equivalence principle — is the starting point for seeing gravity not as a force, but as a property of spacetime geometry.
The principles of equivalence and general covariance, discussed below, are treated as foundational postulates in general relativity. They are not derived from earlier theory, but assumed based on physical reasoning and experimental consistency, just like the two postulates of special relativity.
Historical Context
When Did the Problems Start?
Newton lived from 1642 to 1727. His mechanics assumed absolute space and time, and remained dominant for over two centuries.
Maxwell's equations (1860s) unified electricity and magnetism and predicted that light travels at a constant speed c.
This created a contradiction: Newtonian mechanics expected that velocities add, but Maxwell’s equations predicted a fixed speed of light, regardless of the observer.
Experiments Showing Light's Speed Appears Constant
The key experiment was the Michelson–Morley experiment (1887).
They tried to detect Earth's motion through a supposed "luminiferous aether" — the medium thought to carry light waves, like air carries sound. If the Earth was moving through the aether, then light should appear slightly faster or slower depending on the direction of motion.
They used an interferometer to measure light’s speed in two perpendicular directions.
They ran the experiment at different times of year, as the Earth moved through space around the Sun.
Result: Light always appeared to move at the same speed, regardless of direction or Earth’s motion.
This was deeply puzzling. The implication was that the aether didn’t exist, and that our assumptions about space and time were wrong.
Special Relativity (1905)
Definitions
Event: A point in spacetime labeled by coordinates (t, x, y, z).
Reference frame: A coordinate system used to assign coordinates to events.
Inertial frame: A frame that moves at constant velocity (not accelerating). In such a frame:
A free particle (one not subject to forces) moves in a straight line at constant speed.
This gives a physical meaning to the frame: it’s where Newton’s first law holds.
We define it via particles because motion is what frames do — they describe how matter behaves.
Postulates of Special Relativity
(We say “postulates”, not “axioms” in my post, because general relativity is empirical and geometric: it starts from physically plausible principles and builds mathematical machinery to express them. However, I see postulates similar way as axioms, they are just the starting obvious assumption.)
Principle of Relativity: The laws of physics are the same in all inertial frames.
Constancy of the Speed of Light: Light in a vacuum moves at speed c in all inertial frames, regardless of the motion of the source or observer.
These are based directly on experiments like Michelson–Morley and the consistent success of Maxwell’s theory.
Mathematical Structure
Spacetime: 4D Minkowski space — R⁴ with a non-Euclidean metric.
Metric tensor (ημν): A (0,2) tensor defining the spacetime interval: s² = −c²t² + x² + y² + z².
This is called a metric of signature (−,+,+,+) — one time dimension, three spatial ones.
Lorentz group: The group of linear transformations that preserve the spacetime interval.
Lorentz transformations: The specific maps that relate coordinates between two inertial frames.
Mathematical Components
4-vectors: Vectors in Minkowski space (e.g., 4-position, 4-velocity, 4-momentum). Elements of R⁴ that transform under the Lorentz group.
Worldline: A function mapping proper time to spacetime coordinates — the path a particle traces through spacetime.
Proper time: Time measured by a clock moving with the particle. Defined as: dτ² = dt² − (1/c²)(dx² + dy² + dz²). This is invariant.
Key Results
Time dilation: A moving clock ticks slower relative to a stationary observer.
Length contraction: A moving object is shorter along the direction of motion.
Relativity of simultaneity: Two events simultaneous in one frame may not be simultaneous in another.
Mass-energy equivalence: E = mc² A body's rest mass corresponds to an energy content.
General Relativity (1915)
Geometry Background
In Euclidean geometry, space is flat and the parallel postulate holds.
In the 19th century, Gauss, Bolyai, and Lobachevsky studied non-Euclidean geometry.
Riemann (1854) introduced manifolds and metrics that allow for intrinsic curvature.
Definitions
Manifold: A smooth 4D space that locally looks like R⁴ but may be curved.
Metric tensor gμν(x): Generalization of ημν. Varies with position and defines distances and angles.
Lorentzian manifold: A manifold with a metric of signature (−,+,+,+).
Geodesic: The "straightest possible path" in curved space — the path a free-falling object takes.
Postulates
Equivalence principle: Locally, being in a gravitational field is indistinguishable from being in an accelerating frame. This is a foundational assumption of general relativity.
General covariance: The laws of physics must take the same tensorial form in all coordinate systems. This is also taken as a postulate, not a derived result.
The form of the physical laws must remain the same under any smooth change of coordinates.
It’s not about the values of fields staying the same, but about how the equations look. They must be tensorial equations — because tensors transform in a consistent way under coordinate changes.
In special relativity, only inertial frames (constant velocity) are allowed. Transformations are restricted to Lorentz transformations.
In general relativity, all frames are allowed — rotating, accelerating, etc. So the equations must hold under arbitrary diffeomorphisms (smooth, invertible coordinate changes).
Was the equivalence principle tested?
Yes, and it continues to be tested. Historically, its earliest roots trace to Galileo's experiments and Newton's observation that gravitational mass and inertial mass are equal. But Einstein elevated this to a local physical principle: in a small region of spacetime, you cannot distinguish between free fall and uniform acceleration.
Key experimental confirmations include:
Eötvös experiment (1889): Showed that different materials fall with the same acceleration to within tiny precision.
Lunar Laser Ranging (ongoing since 1969): Confirms Earth and Moon fall toward the Sun at the same rate, despite being made of different stuff.
Modern satellite tests (like MICROSCOPE, 2017): Confirm the principle to within 1 part in 10¹⁴.
Mathematical Structure
Christoffel symbols (Γ): Coefficients defining how vectors change when moved in curved space. Γ^σ_{μν} = ½ g^{σλ}(∂μ gνλ + ∂ν gμλ − ∂λ gμν)
Covariant derivative: Derivative that respects curvature.
Riemann tensor (R^ρ_{σμν}): Measures how parallel transport around a loop changes a vector.
Ricci tensor (Rμν): Contraction of the Riemann tensor.
Ricci scalar (R): Trace of the Ricci tensor.
Einstein tensor (Gμν): Gμν = Rμν − ½ R gμν
Stress-energy tensor (Tμν): Describes matter, momentum, and energy content of spacetime.
Einstein Field Equations
Gμν = (8πG / c⁴) Tμν
Matter and energy determine how spacetime curves. That curvature determines how objects move.
It’s the central equation of general relativity — the one that took Einstein around 8 years to develop (from 1907 to 1915).
Left side: Geometry of spacetime
The Einstein tensor - made from the metric and its derivatives (involving curvature). It encodes how spacetime is shaped.
Built from the Ricci tensor and Ricci scalar.
It satisfies the conservation law
Right side: Energy and momentum
The stress-energy tensor - represents the distribution of matter and energy (mass, momentum, pressure, energy density...).
G: Newton’s gravitational constant.
c: Speed of light.
Key Results
Gravitational time dilation: Time runs slower in stronger gravitational fields.
Deflection of light: Light curves around massive objects.
Black holes: Solutions (like Schwarzschild) predict regions of infinite curvature.
Expanding universe: Friedmann solutions describe dynamic, expanding spacetimes.
Gravitational waves: Ripples in spacetime, confirmed in 2015.
More notes
CPT symmetry
CPT symmetry (charge, parity, time reversal) is a principle from quantum field theory, not general relativity. However, special relativity is built into QFT, so Lorentz invariance is essential to CPT symmetry being well-defined. They’re related, but CPT is not a direct consequence of relativity.
Where does spacetime unity come from?
In special relativity, it follows from:
The postulates themselves — especially the second:
Postulate 2: The speed of light is the same in all inertial frames.
This forces us to abandon the idea of absolute simultaneity, which means time becomes observer-dependent — just like position already was in Newtonian mechanics.
That leads to the realization:
To preserve the laws of physics (and especially the constancy of light speed), space and time must be treated together.
Mathematically, this is formalized through:
The invariant spacetime interval:
s² = -c²t² + x² + y² + z²
This formula treats time and space as components of a 4-dimensional geometry. You can't separate them without breaking the symmetry of the theory.
Why is E = mc² the most famous result?
It’s saying that mass is not just inertia, it’s frozen energy. It can be transformed.
Several reasons why this is the most famous
It's simple, elegant, and universal.
It connects three fundamental concepts: energy (E), mass (m), and the speed of light (c), with just an equals sign and a square.It broke old intuition.
Before this, mass and energy were thought to be separate, conserved quantities. Einstein showed they are interchangeable, mass is energy, and vice versa.It had enormous implications.
It explained how the Sun produces energy (via nuclear fusion, converting mass into light and heat).
It laid the foundation for nuclear energy and atomic weapons.
It fundamentally changed how physicists think about conservation laws.
It became a symbol of Einstein himself.
Popular science writers, textbooks, and media latched onto it, and it entered cultural consciousness far beyond physics.
Symmetry, groups:
What does it mean to “break the symmetry of the theory”?
In special relativity, the symmetry group of spacetime is the Lorentz group (plus translations, forming the Poincaré group). This group consists of all transformations that preserve the spacetime interval:
s² = -c²t² + x² + y² + z²
These include:
Rotations in space (SO(3))
Boosts (changes of inertial frame at constant relative velocity)
The fact that this interval is preserved under these transformations means that space and time are unified: they mix into each other under the group action (boosts). So:
If you treated space and time as fundamentally separate, you’d be privileging one direction (say, time) over others.
But the Lorentz group mixes space and time, just like SO(3) mixes x, y, z in Euclidean space.
‘frozen’ energy is a cool way to
put it.