Gravity, Spacetime, and Motion Fundamentals

Posted on Nov 19, 2024 in Physics

Gravity & Spacetime

Laws of Physics are determined by:

1. Size: Big & Small

2. Speed: Fast & Slow

• Larger than 10^-9m (big)
• Smaller than 10^-9m (small)
• Comparable to 3×10⁸m/s (fast)
• Slower than 3×10⁸m/s (slow)

• Classical Mechanics (big & slow)
• Quantum Mechanics (small & slow)
• Relativistic Mechanics (fast & large)
• Quantum Field Theory (small & fast)

General Relativity

• Space & time form a 4D “fabric”
  • Can be curved by mass & energy

Gravitation is not a force but a curvature of spacetime.

• The Einstein Field Equations
  • Shows relationship between spacetime curvature & distribution of mass & energy

• G_ij: Curvature
• T_ij: Mass/Energy
• c: Speed of Light
• G: Newtonian Gravitation Constant
• G_ij = (8πG/c⁴) (T_ij)
• c: 3×10⁸ m/s, G: 6.67×10^-11 m³Kg^-1s^-2

The Curvature of Spacetime

• Spacetime can be analogized as 2D sheet
• Mass/Energy distort the sheet into 3D
• Minkowski Spacetime is flat: t = d/c
• t_curve > t_flat
• The more gravity, the slower time passes

The Chandrasekhar Limit (1.44M_☉)

• When a star runs out of fuel, gravity pulls the remaining mass in on itself.
• Fate of a star depends on initial mass.
• If mass is not M < 1.44M_☉ to our sun, it will become a red giant then a white dwarf.

The Tolman-Oppenheimer-Volkoff Limit

• If 1.5M_☉ < M < 3.0M_☉ it will turn into a white dwarf and keep shrinking.
• Then stopped by quantum degeneracy pressure.
• Results in neutron star.
• If mass of star is beyond TOV, result is black hole.
• Occurs when a mass M is compressed to have a radius of 2GM/c².

• Radius of compressed volume is called Schwarzschild Radius (R_s).

Quantum Mechanics & Gravity

• General Relativity (large objects)
• Quantum Mechanics (small objects)
• If small object has a large gravity
  • New Theory is required
  • Not yet discovered

Distance & Displacement

Distance

• The linear length of path taken between two points in spacetime.
• Scalar quantity (no direction).

Geodesic

• The path of shortest length taken between points.
• Minkowski Spacetime: Straight Line (with direction).
• Sphere: Great Circle – any circle whose centre is at the centre of sphere (no latitude but all longitudes) (with direction).

Displacement

• The length of, and direction of travel along, the geodesic.
• Vector Quantity.

Symbols for Distance & Displacement

• Distance and Displacement in 2D must account for magnitude and direction.
• In order to compute →d_TOT, →d₁ and →d₂ must be broken down into their horizontal and vertical components (→d_1x & →d_1y and →d_2x & →d_2y).

Speed & Velocity

Speed

• The rate of change of distance with respect to time.
• Scalar Quantity.

Average Speed

• The quotient of the total distance and the total time.
• The constant rate at which you would need to travel to cover the required distance in the required time.

<v> = d/t

<v> = d_TOT/t_TOT

<v> = (v₁ + v₂)/2

Velocity (→v)

• The rate of change of displacement with respect to time.
• Vector quantity.
• Symbol is →v.

Average Velocity

• The constant rate at which you would need to travel to cover the required displacement in the required time and the associated direction.

<→v> = →d_TOT / t_TOT

<→v> = Δ→d / Δt

<→v> = (→v₁ + →v₂)/2

The Addition of Velocities

• Velocities are added in the same manner displacements

(or any other vector)

vTOT=v₁+v₂+…vn

• If velocities are perpendicular:

vTOT^2=(v₁)^2+(v₂)^2 & tan=v₂/v₁

A Gedanken (thought) experiment about c

• “Einstein-Rosen Bridge”

• “Transversable Wormhole”

• Wormholes require negative energy to be kept open

Acceleration

Acceleration

• The rate of change of velocity with respect to time

• Vector quantity

• Symbol is a

a=v/t

• Measured in m/s2

• Can be measured in g’s — the number of times the

acceleration is of Earth’s acceleration due to gravity

9.8m/s2

d=do+vot+1/2 at^2

v^2-vo^2=(2a)d

= v₁+v₂/2

dtot=()(tTOT) or d=()t

Average Acceleration

=v/t

Positive & Negative Acceleration Consequences

• Positive acceleration does not always

make you go faster

• Negative acceleration does not always

make you go slower