Why do all masses fall at same rate? All normal forces (e.g. electrical, friction, elastic...) don't produce same accn in all bodies.
F = ma = mg so a = g
Are we really sure the m's are the same? This concerned Newton
The first m (inertial mass mI) measures how hard things are to accelerate (2nd. law), the second (gravitational mass mG) measures gravitational force
F = mIa
F = mGg
Pseudo-forces (e.g. centrifugal force) behave the same way
F = mv2 = ma
R
so all bodies undergo same centrifugal accn.
Maybe gravity is somehow a fictitious force (?!?!?!?)
mI = mG
F = mIa = mIg
so a = g only if the "inertial mass" the gravitational mass. Can demonstrate this is true to 1 part in 1012 (Eötvos experiment).
Special relativity said you cannot do an experiment to decide if you are moving. General says that you cannot do an experiment to distinguish between a gravitational field and an acceleration (!!!!!!!!!)
For example:
Suppose you are in a stationary elevator, and a bullet is shot horizontally, it will fall due to gravity..
Suppose you are in an accelerating elevator, and a bullet is shot horizontally, it will appear to fall..
You cannot distinguish the two.
Suppose you are in an accelerating elevator, and a beam of light is shot horizontally, it will appear to fall..
Suppose you are in a stationary elevator, and a beam of light is shot horizontally, it will fall..
You cannot distinguish the two. Light gets affected by gravity?
General relativity:
Handles frames which are accelerating w.r.t. each other.
What is a straight line?
which is the straight line?
General relativity: metrics
Physics should not depend on the frame of reference: e.g. which way is up?
Principle of General Covariance: physical laws are the same in any frame.
(e.g. Newton;s are not, since they aren't the same in an accelerating frame.
Metric is distance in terms of coords: in Cartesians:
Δs²=Δx²+Δy²+Δz²
In spherical polars:
Δs²=Δr² + r²(Δϑ² + sin²θΔφ²
Δs² must be the same. Can include time via special rel:
Δτ²=Δt² - (Δx²+Δy²+Δz²)/c²
where τ is the proper time. e.g. for a moving body, this could be the interval between creation and decay of a particle, the (invariant) lifetime.
This contains (e.g.) time dilation: if the particle is travelling in some frame with vel
v = Δr/Δt (t is the time measured in the frame then
Δτ²=Δt² - v²(Δt²)/c²
so immediately
Δt² = Δτ²/(1-v²/c²)½
Note that this implies that proper time can be real or imaginary: more specifically if we have two events
Δτ² < 0 the events have a space-like separation
Δτ²= 0 the events have a light-like separation
Δτ² > 0 the events have a time-like separation
Defines "metric tensor" for us:
in 3-D, the distance between 2 points $
\color{red}{\Delta s^2 }$
cannot depend on the coord. system. Suppose in one system we have
$$
\color{red}{
\Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 }
$$
and we write $
\color{red}{x = x\left( {x^1 ,x^2 ,x^3 } \right)}$ etc..
Then in this case $$
\color{red}{
\begin{array}{l}
x = r\sin \left( \varphi \right) \\
y = r\cos \left( \varphi \right) \\
\end{array}}
$$
Can now find $$
\color{red}{
\Delta x = \frac{{\partial x}}{{\partial x^1 }}\Delta x^1 + \frac{{\partial x}}{{\partial x^2 }}\Delta x^2 + \frac{{\partial x}}{{\partial x^3 }}\Delta x^3 }
$$
etc. Then we can write$$
\color{red}{
\begin{array}{l}
\Delta s^2 = \left( {\frac{{\partial x}}{{\partial x^1 }}\Delta x^1 + \frac{{\partial x}}{{\partial x^2 }}\Delta x^2 + \frac{{\partial x}}{{\partial x^3 }}\Delta x^3 } \right)^2 + ..... \\
= \sum\limits_{}^{} {\sum\limits_{}^{} {g_{\mu \nu } \left( {x^1 ,x^2 ,x^3 } \right)\Delta x^\mu \Delta x^\nu } } = g_{\mu \nu } \Delta x^\mu \Delta x^\nu \\
\end{array}}
$$
Einstein summation convention: repeated indices are summed over.
Note:
$\Delta s^2 $ is invariant (Doesn't depend on the coord system or "frame of reference"
$\Delta x^1 $ etc do depend on frame
in general $
\color{red}{g_{\mu \nu } }$
depends on the location. Most (all!) of the cases we deal with will have a very simple dependence on space or time.
in 3-D (4-D) there are 9 (16) terms in $
\color{red}{g_{\mu \nu } }$
because of symmetry $$
\color{red}{
g_{\mu \nu } = g_{\nu \mu } }
$$
so 6 (10) of these are indep.
Note that we can have spaces with no metric: putting in metric gives geo"metry".
in special relativity
$$\color{red}{
g_{\mu \nu } = \left[ {\begin{array}{*{20}c}
1 & 0 & 0 & 0 \\
0 & { - \frac{1}{{c^2 }}} & 0 & 0 \\
0 & 0 & { - \frac{1}{{c^2 }}} & 0 \\
0 & 0 & 0 & { - \frac{1}{{c^2 }}} \\
\end{array}} \right]}
$$
In fact we often define c = 1. Note the space-like terms come with negative sign.
In GR we will have a more general metric.
Dynamics:
Why bother?
A Body continues at rest or in a state of uniform motion unless acted on by
a force.
Uniform motion means in a straight line.
A geodesic in Euclidean space ≡ straight line ≡ shortest path.
Geodesic: shortest time path between 2 points --> straight lines in a flat space
Geodesic is "extremal path": in GR the proper time is a max. Massive bodies follow timelike geodesics. Planets are actually moving in "straight" lines in a curved space...
Can either say:
There is a force called gravity which acts on all energies (and hence attracts light)
There is no such thing as gravity, it's just that masses distort space-time in their neighbourhood
Either way, don't jump off tall buildings: you can be just as dead in a curved space!
"Lenses extend unwish through curving wherewhon till unwish returns on its unself"
e.e.cummings
Now we need to explain exactly what we mean by curved spaces
