Mechanics and Relativity la Relativity

Mechanics and Relativity, la Relativity, Part 1 of the course. The course consist 2 parts, but I followed only part 1 (in Period 1a), at the University of Groningen, year 1, 2020-2021, Lecture notes and book summarized.
Mechanics and Relativity, la Relativity, Deel 1 van het vak. Het vak bestaat uit 2 delen, ik heb alleen het eerste deel gevolgd (periode 1a) aan de rijksuniversiteit van Groningen, jaar 1, 2020-2021. Hoorcolleges aantekeningen en boek samengevat. In het Engels want dat is alles.

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Lecture 1
1.1
History of Relativity:
• Theory of Relativity (1905-1915): c 6= 0
• A. Einstein (1905): The laws of physics are the same inside a laboratory moving at a constant
velocity as they are in a laboratory at rest.
• Theory of quantum-mechanics (1925):h 6= 0
• ether: hypothetical medium;light. Physics not able to detect. Einstein came with proposed
solution −
→ conclusion: Galilean transformation equations incorrect high velocity’s
1.2
Einstein Equations:
• G(µ · v) + g(µ · v) = 8 · π · T(µ · v)
• curvature=energy/mass
• Made predictions that have been tested that do not hold in General Relativity: relativistic
corrections to motion of planets around the sun, bending of light etc.
1.3
Principles
• The principle of Relativity: the laws of physics are the same in all inertial reference frames:
– invariant under translations, rotations, boosts.
– Speed of light is constant, frame invariant and not infinite.
– consequences: time 6= absolute.
– Not proven, not disporven.
• equivalence principle: inertial mass= gravitational mass (Galileo Galilei, Keppler, Einstein).
• General Relativity: Newtonian Gravity+special Relativity (SR(Einstein))
• The principle of Relativity:
– 1916: ”Die Grundlage der Allgemeine Relativit¨
atstheorie” by Albert Einstein.
– Accelerated observer? Mathematically more complicated.
– Space-time is equated with energy (density).
– The laws of physics for observers: gravitational field (curved space) = the law of physics
for accelerated observer.
– converted notion onto testable hypothesis.
– all experimental evidence is consistent with it. Need to made more precise: see 1.5.
• The principle of Newtonian Gravity:
– standard orientation: axes same direction in space, Other (primed) Frame, moves
along x-axis relative to Home (unprimed) frame.
– Newton’s hypothesis: time is universal, and absolute.
– Time is independent of reference frame;
1.4
Galilean transformations:
Galilean transformations: view Newtonian concept. How relate Home and other frame? Move-
ment along y-axis: ~
r0(t0) = ~
r(t) − ~
βt
β =velocity Other frame relative Home frame.
In general:
Galilean transformation equations: position of object at given time t0 in the Other frame
if we know position t = t0 in home frame.
Galilean velocity transformation equations: velocity of object when velocity in home
frame is familiar:
Galilean transformation equations
Galilean velocity transformation equations
t0 = t
x0 = x − βt
v0 = v
x − βt
y0 = y
v0 = v
z0 = z
v0 = v
acceleration: derivative Galilean velocity transformation equations → a0 = a → observers
agree acceleration independent position and velocity
limitations: t = t0 only true when ~
v c and ~
β c
1.4.1
Example:
←
− Alice on boat, throws ball in air.
→ Bob observes Alice in boat.
Boat is moving @β Alice&Bob does not agree on position ball in respective frames: ~
r 6= ~
Agree on acceleration ball: Newton’s law: F − m · a holds.
Law of physics invariant under Galilean transformations.
1.5
Events and coordinates:
”Laws of physics are the same instead a laboratory that moves with a constant velocity and
inside a laboratory at rest”.(A. Einstein)
Laboratory: Provide a men of measuring the spacetime coordinates of event.
CAN NOT distinguish a reference frame ”moving at constant velocity from one at rest”.
relative: depends on reference frame.
Events: Something that happens @well-defined place & time. labelled by time t
worldline: displays how the particle’s position varies with time, the set of all events occurring
along the path of an object. → The worldline of a particle: A series of events (f.e. the motion
of a particle).
spacetime coordinates: set 4 numbers, locate event in space& time.
spacetime Diagram: graph for depict the coordinates of events. Always position vs. time.
origin event: the event on the t-axis, which is the worldline of the Homeframe.
Right handed Cartesian Coordinate system: Coordinatesystem (3d) lattice with origin.
vector: ~
r = (x, y, z)
attaching a clock @ each point of our lattice?−
→ An event registered using (t,~r)
x 












−2
r1 =
3 , ~
3 , ~
∆r =
−

1 


2 =  2 








Length is same, also when you change direction.
↑ you see that ~
r2 + ~
∆r = ~
r1 So, ~
∆r = ~
r1 − ~
1.6
General vector calculus rules:
ax
bx
cx
bx ± cx
addition and subtraction: ~a = ~b + ~
c →
by ± cy








bz ± cz