ASTR Lecture Notes 10/17

1905: Special Relativity - Addressing the unique nature of the speed of light.
1915: General Relativity - Redefining gravity based on new understandings.

Relativity of Simultaneity: Events may seem simultaneous to one observer while not to another.
Time Dilation: The faster a clock moves relative to an observer, the slower it ticks.
Length Contraction: The faster an object moves, the smaller it appears to an observer.
Mass Increase: The mass of an object increases as its speed approaches the speed of light.
Ultimate Speed Limit: No object with non-zero mass can exceed the speed of light.
Mass-Energy Equivalence: Expressed as $E=mc^2$, denotes the convertibility of mass and energy.
New Physics: Newtonian physics are approximations valid at low velocities (i.e., $v << c$).

Essential Concept: The effects of gravity are indistinguishable from the effects of acceleration. Thus, the formula for force, $F=ma$, relates to relativity.
Implications take time to understand fully.

Free-floating objects follow the straightest possible path through spacetime, while acceleration or gravity leads to curved paths.
Newton's View: Described a phenomenon of action at a distance.
Einstein's View: Defined gravity as movement along a curved path in spacetime, leading to the understanding that gravity distorts spacetime itself.

The difficulty in picturing curved spacetime. Simplifying to a 2D curve, navigating north-south and east-west, provides insight.
Sphere Concept: A sphere provides a model of a curved surface, illustrating that the shortest path, or great circle, is not a straight line.

Illustrates that gravity creates curves; objects follow the shortest path, which in curved space can mean moving along a curve.
Noting limitations: This analogy simplifies multi-dimensional realities and neglects the time aspect.
In a 3D context, mass influences how spacetime curves.

Objects naturally aim to travel the shortest possible path through spacetime. In a curved spacetime, their paths become curved.
Einstein’s Proposition: Light and matter both follow these paths through spacetime.

An elevator accelerating upwards at $1 ext{ m/s}^2$. The light beam observed across a moving elevator will not follow a straight path from the perspective inside the elevator.
Visualization times for different t values (t = 0, 1, 2, 3).

The Equivalence Principle states that one cannot distinguish between acceleration and gravitational force within an elevator.
If acceleration can bend light rays, gravity must also bend light rays.
Light rays passing near massive objects like the Sun will shift from their apparent positions:
- The displacement of stars near the Sun, observed during the total solar eclipse of May 29, 1919, shifts by approximately $1.75$ arc-seconds.

Sir Arthur Eddington tested Einstein’s theory during a solar eclipse, confirming light bending around massive bodies.
Observations conducted from two locations: northern Brazil and the west coast of Africa.

Einstein Cross: Formation of multiple images of a distant object due to lensing.
Einstein Ring: A more structured manifestation of gravitational lensing that creates a ring-like structure from multiple lensed images.
Brightening occurs due to lensing as the total light adds up from multiple images.

Lensing can identify exoplanets; as of early 2025, 237 exoplanets had been discovered using this method.
Disadvantage: Once observed, data cannot be followed up, as the phenomenon is transient.

Clocks in stronger gravitational fields tick slower compared to those in weaker fields.
GPS systems must compensate for this dilation as they exist in both weak and strong gravitational conditions in relation to Earth.
Example: Comparisons of atomic clocks in different gravitational fields reveal variances in time measurement.

Newton's laws become inaccurate near massive bodies and at speeds close to light:
- Mercury’s elliptical orbit illustrates precession, a phenomenon only explained through Einstein’s general relativity.

Spacetime is four-dimensional.
Mass causes spacetime to curve and bend the trajectories of light rays.
Gravity is explained through the geometry of curved spacetime.
Time slows down significantly under strong gravitational influence, with implications for both practical applications and theoretical explorations in physics.

Concepts span a plethora of advanced topics, including black holes, gravitational waves, and hypothetical warp drives.

Reflections on the 20th century emphasize its advancements in science and technology, with Einstein representing a pivotal symbol of scientific progress.

Analyzing a binary system reveals a mass of $6 M_{sun}$ for a hidden star, considering a nearby supernova event influences its mass classification:
- $ullet$ A white dwarf is a remnant for less than $7 M_{sun}$.
- $ullet$ A neutron star is the remnant core post-supernova for massive stellar cores.
- $ullet$ A black hole is formed when collapsing cores exceed specific mass limits (approx. $2.5-3 M_{sun}$).

Stellar evolution progresses from protostellar nebula through stages of different mass classifications ending in various stellar remnants:
- high-mass main sequence star → red supergiant → supernova → neutron star/black hole.

Core pressures countering gravitational collapse include:
- Gas pressure (via fusion in main sequence stars)
- Electron degeneracy pressure (maximum mass $M < 1.4 M_{sun}$)
- Neutron degeneracy pressure (maximum mass $M < 3 M_{sun}$)
Without counteracting forces, gravity overwhelms, resulting in black hole formation.

Singularity Concept: In the rubber sheet analogy, a black hole represents an extreme point of curvature where mass collapses to an infinitesimal point.
Black Hole Definition:
- Escape velocity ($V_{escape}$) exceeds $c$ (the speed of light).
- The unique radius where $V_{escape} = c$ is termed the event horizon or Schwarzschild radius.

Event horizon radius calculations for various black hole masses yield:
- For $M = 1 M{sun}$, $r{EH} = 3 km$.
- For $M = 3 M{sun}$, $r{EH} = 9 km$.
- For $M = 200 M{sun}$, $r{EH} = 600 km$.