Lecture 9: Relativity

Interest in Interstellar Travel: Hollywood's portrayal vs. scientific reality.
Pioneer 10 (1972):
- First spacecraft from Earth to reach Jupiter (~628 million km away).
- Distance to Nearest Star: Proxima Centauri is 4.25 light-years (ly) away.
- Estimated travel time at current speeds: ~2 million years to Proxima Centauri.
- Aldebaran's Distance: 68 ly away.
Current Spacecraft: Voyager 1 and 2 will not approach any star within 0.75 ly.
Speed record held by the NASA Parker Solar Probe: 690,000 km/h (430,000 mph) or 191 km/s.
- Example: Would travel from NYC to Tokyo in 1 minute, but reaching Proxima Centauri would take 4250 years.

Speed of Light (c): Universal speed limit at c = 3 x 10^8 m/s.
Special Theory of Relativity: A major obstacle for interstellar travel; speeds required exceed 10,000 ly at 0.1c.

Relative Motion: Motion is only meaningful relative to a frame of reference.
- Example: Plane traveling westward is stationary in Earth's frame while Earth rotates.
Postulates of Special Relativity:
1. The laws of nature are the same for all observers.
2. The speed of light in free space is constant.
Consequences:
- Loss of Simultaneity: Events simultaneous in one frame may not be in another.
- Time Dilation: Moving clocks run slower relative to stationary observers.
- Length Contraction: Moving objects appear shorter in the direction of motion.
- Example of Loss of Simultaneity:
- Two observers see flashes of light; one sees them simultaneously while the other sees one before the other.

At high speeds, the addition of velocities requires adjustments:
- If observer A sees B moving at speed $v1$ and B sees C at speed $v2$, the observed speed of C is not simply $v1 + v2$ but recalculated using relativistic formulas.

Example of Time Dilation:
- At $v = 0.87c$: 2 Earth years = spacecraft year.
- At $v = 0.995c$: 10 Earth years = spacecraft year.

Definition: The factor quantifying time dilation and length contraction:
- Formalism: Derived from simple geometry of spacetime changes.
- Denote:
- Proper time in the moving frame as $t_0$.
- Relative time in another frame as $t$.

Core Idea: Gravity results from the curvature of spacetime; mass causes spacetime to curve, resulting in gravitation.
Spacetime: Four-dimensional continuity including three spatial directions (x, y, z) and time.
Consequences of General Relativity:
- Gravitational Time Dilation: Clocks nearest a gravitational source run slower.
- Gravitational Lensing: Light paths bend around massive objects.

Parameters in GR are used to model gravity:
- $𝛾$: Represents space curvature from unit mass.
- $β$: Represents mass nonlinearity.
- General relativity predictions have been quantified, validating GR vs. Newtonian physics:
- GR values: $𝛾, β < 1e-5$.

Einstein's Proposed Tests:
1. Perihelion precession of Mercury's orbit.
2. Deflection of light by the Sun.
3. Gravitational redshift of light.
Irwin Shapiro's Test:
- Time delay in radar signals traveling near the Sun.
Observational Evidence:
- Gravitational waves have provided additional supportive evidence for GR.

Perihelion Precession:
- Observed advance of Mercury's orbit by 43”/century, confirmed by GR.
Light Deflection:
- Prediction from GR: 1.75 arcseconds, double Newtonian prediction of 0.875 arcseconds.
- Confirmed by the 1919 solar eclipse.
Gravitational Redshift:
- Light from massive bodies shifts towards red, confirmed with data from atomic clocks.

Predicted by GR; ripples in spacetime triggered by massive objects.
- Indirect evidence from binary pulsars (Taylor and Hulse, 1974).
- Direct detection by LIGO, confirmed the merger of black holes (2017).

Concept: Hypothetical cosmic shortcuts, not yet proven plausible due to extreme energy requirements.
Theoretical Concerns:
- Stability and causality issues in creation of and travel through wormholes.

Example of energy needed to travel:
- Required to accelerate a starship to 10% of c takes: E= rac{1}{2}mv^2 = 4.5 imes 10^{22} ext{ Joules}.
- This amount is roughly 100 times the world’s entire annual energy consumption.
Theoretical Limits: As speed approaches light, achieving acceleration becomes more difficult.

Tides: Result from gravitational interactions, notably with the Moon and Sun.
- Strength of tides is proportional to rac{M}{d^3} where M = mass of the tide-raising body and d = distance to it.
- Tidal effects alter planetary rotations and orbits over time.

Example: Earth's rotation slows at a rate of 2 ms/century due to tidal friction.
Synchronous Rotation: The Moon’s rotation period matches its orbital period due to tidal locking.

Definition of Cognitive Bias: A tendency to think in specific ways that can lead to systematic deviations from rationality.
- Examples of Cognitive Biases:
- Anchoring Bias: Relying too much on initial information.
- Confirmation Bias: Focusing only on confirming evidence.
- Availability Bias: Overestimating likely outcomes based on recent events.

Definition of a Fallacy: Incorrect reasoning in argumentation.
- Examples:
- Ad Hominem: Focusing on the individual instead of the argument.
- Begging the Question: Assuming the premise in the conclusion.
- Red Herring: Diverting attention from the main issue.

UFO/UAP Reports: Majority are debunked or explained as natural phenomena or misidentifications.
Conspiracy Theories: Often arise from biases; claims of cover-ups lack substantial evidence.
Scientific Approach: Critical thinking and peer review are essential for discerning truth in UFO claims.