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General relativity: intro

General relativity: intro

Why do all masses fall at same rate? All normal forces (e.g. electrical, friction, elastic...) don't produce same accn in all bodies.

\color{red}{ F = m_I a = m_G g \Rightarrow a = g}

Are we really sure the m's are the same? This concerned Newton
The first m (inertial mass m_I) measures how hard things are to accelerate (2nd. law), the second (gravitational mass m_G) measures gravitational force
F = m_Ia F = m_Gg
Pseudo-forces (e.g. centrifugal force) behave the same way. $$ \color{red}{ F = \frac{{mv^2 }}{R} = ma} $$
so all bodies undergo same centrifugal accn.

Maybe gravity is somehow a fictitious force (?!?!?!?)

\color{red}{ m_I \equiv m_G }

so a = g only if the "inertial mass" is 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.

For example:

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?
This is the Equivalence Principle

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:

\color{red}{ \Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 }

In spherical polars:

\color{red}{ \Delta s^2 = \Delta r^2 + r^2 \left( {\Delta \theta ^2 + \sin ^2 \left( \theta \right)\Delta \varphi ^2 } \right)}

Δs² must be the same. Can include time via special rel:

\color{red}{ \Delta \tau ^2 = \Delta t^2 - \frac{1}{{c^2 }}\left( {\Delta x^2 + \Delta y^2 + \Delta z^2 } \right)}

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

\color{red}{ \Delta \tau ^2 = \Delta t^2 - \frac{{v^2 }}{{c^2 }}\Delta t^2 }

so immediately

\color{red}{ \Delta t = \frac{{\Delta \tau }}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}}

Note that

\color{red}{ \Delta \tau ^2 = \Delta t^2 - \frac{{\Delta r^2 }}{{c^2 }}}

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.. e.g. for polar coords $$ \color{red}{ \begin{array}{l} x = r\sin \left( \varphi \right) \\ y = r\cos \left( \varphi \right) \\ \end{array}} $$ 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.
"Euclidean Metric" $$ \color{red}{ g_{\mu \nu } = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}} \right] = g_{ij} } $$ gives us Euclidean geometry

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