Special vs General Relativity

Date:: 25 Sep 2019

What is the problem ?

Gravity is two objects pulling each other
Einstein thinks the “pulling” is actually achieved by
- Object has mass, this mass causes space to deform
- The deform causes two object to move closer, as if they are pulling each other
In Special Relativity, we don’t consider gravity
In General Relativity, we do consider gravity
Relativity studys the Geometry of Spacetime

Derivation

Step 1 When we study geometry, we must define Coordinates and Metric, then we can talk about distance.

\[\begin{split}\begin{align*} & \text{Coordinates: Alice stand at } x=3 \text{, Bob } x=5\\ & \text{Metric: They agree that each unit length }g=5m\\ & \text{Then their distance }\Delta s = g \cdot \Delta x = 5m\cdot 2=10m\\ & \text{Consider squared length of a small distance }(ds)^2\\ & \text{Notation: } ds^2=(ds)^2\\ & \text{2D Euclidean Coordinates: } ds^2=dx^2 + dy^2\\ & \text{2D Polar Coordinates: } ds^2= dr^2 + r^2 d\theta^2\\ & \text{Write in General form: } ds^2= g_{ab}dx^{(a)}dx^{(b)}\\ & g_{ab} \text{ is called Metric Tensor and Einstein summation convention is used}\\ & \text{Then:}\\ & \text{2D Euclidean: } x^{(0)}=x, x^{(1)}=y\text{ and } g_{00}=g_{11}=1, g_{01}=g_{10}=0\\ & \text{2D Polar: } x^{(0)}=r, x^{(1)}=\theta\text{ and } g_{00}=1, g_{11}=r^2, g_{01}=g_{10}=0 \\ & \text{SR = Special Relativity, GR = General Relativity}\\ & \text{4D Spacetime(SR): Ignore mass deforming space} \\ & \;\;\;\; \boxed{ds^2 = -dt^2+dx^2+dy^2+dz^2}\\ & \text{4D Spacetime(SR): } x^{(0)}=t, x^{(1)}=x, x^{(2)}=y, x^{(3)}=z\\ & \text{4D Spacetime(SR): } g=g_{\text{Minkowski}}= \left(\array{ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 }\right) \\ & \text{4D Spacetime(GR): Consider mass M deforming space} \\ & \;\;\;\; \boxed{ds^2 = -(1-{r_s\over r}) dt^2+(1-{r_s\over r})^{-1} dr^2 +r^2 d\theta^2+ r^2 \sin^2 \theta d\phi^2 }\\ & \text{Where } r_s=2M \text{ is called Schwarzschild radius}\\ & \text{4D Spacetime(GR): } x^{(0)}=t, x^{(1)}=r, x^{(2)}=\theta, x^{(3)}=\phi \\ & \text{4D Spacetime(GR): } g=g_{\text{Schwarzschild}} \\ \end{align*}\end{split}\]

Step 2 Some definitions

\[\begin{split}\begin{align*} & \textbf{I. Some definitions: SR \& GR }\\ & \text{Inner Product: } A\cdot B= g_{ab}A^{(a)}B^{(b)} \\ & \text{Proper Time: } d\tau :=\sqrt{-ds^2} \\ & \text{Four Velocity: } u^{(a)}:={dx^{(a)}\over d\tau}\\ & \Rightarrow u^2=u\cdot u = g_{ab}{dx^{(a)} \over d\tau}{dx^{(b)} \over d\tau} = {ds^2\over -ds^2} =-1\\ & \text{Four Momentum: } p:=mu\Rightarrow p^2=-m \;\;\;\; =0 \text{ for Photon}\\ & \text{Relativistic Energy: } E:=-g_{00} p^{(0)}\\ & \text{Relativistic Momentum: }\vec{p}:=(p^{(1)},p^{(2)},p^{(3)})\\ & \textbf{II. SR}\\ & \text{Velocity: } \vec{v}=({dx\over dt},{dy\over dt},{dz\over dt})\;\;\;\; |\vec{v}|\le c\equiv1\\ & \Rightarrow ds^2 = -dt^2+dx^2+dy^2+dz^2 = (-1+\vec{v}^2)dt^2\\ & \text{Proper Time: } d\tau :=\sqrt{-ds^2}=\sqrt{1-\vec{v}^2}dt\\ & \text{Relativistic Energy: } E:=-(-1)p^{(0)}=m{dt\over d\tau}=m {1\over \sqrt{1-\vec{v}^2}}\\ & \text{Relativistic Momentum: }\vec{p}=m {\vec{v}\over \sqrt{1-\vec{v}^2}}\\ & \textbf{III. SR Photon}\\ & \text{Four Momentum: }p=(E,Ev_x,Ev_y,Ev_z)\\ & \text{Relativistic Energy: } E_{\text{Frame of source}}\equiv E_0={1\over \lambda_0}\\ & \;\;\;\; \text{(Refer to Lorentz Transformation for below:)}\\ & \;\;\;\; v_x=1,v_y=v_z=0 \Rightarrow(E,E,0,0)=\Lambda (E_0,E_0,0,0)\\ & \;\;\;\; \Rightarrow \boxed{ E=E_0 \gamma-E_0 \gamma v \;\;\;\; \text{Doppler effect} }\\ & \textbf{IV. GR}\\ & \text{Proper Time: Choose coordinate such that } \\ & \;\;\;\; dx^{(1)}=dx^{(2)}=dx^{(3)}=0\Rightarrow d\tau :=\sqrt{-g_{00} }dt =\sqrt{(1-{r_s\over r}) } dt \\ & \text{Application: Gravitational Redshift} \\ & \;\;\;\; \lambda =cT,c=1\Rightarrow {\lambda_{\text{Receiver}} \over \lambda_{\text{Emitter}}} = {\Delta \tau_R \over \Delta \tau_E}={\sqrt{1-r_s/ r_R } \over \sqrt{1-r_s/ r_E }} \\ & \;\;\;\; \approx 1-{M\over r_R}+ {M\over r_E}\overset{\Delta r\ll r_E}{\approx} 1+{M\over r_E^2}\Delta r =1+g\Delta r\\ \end{align*}\end{split}\]

Step 3 Equation of Motion: the Geodesic Equation

Also base on Variational Principle, same as Newton’s equation of motion and Shrodinger Equation
Specifically, we will maximize proper time

\[\begin{split}\begin{align*} & \text{Geodesic: (1) Path with longest proper time. OR (2) Straight line}\\ & \text{Parameterize the line with one varibale: }l\in [0,1]\\ & \text{Notation: } \dot{x}={dx\over dl}\\ & \tau = \int \sqrt{-ds^2}= \int_0^1 \sqrt{-g_{ab}\dot{x}^{(a)}\dot{x}^{(b)}} dl\equiv \int_0^1 L(x,\dot{x};l) dl\\ & \text{Note that } d\tau = L dl\\ & \text{To maximize }\tau \text{, use Euler-Lagrange Equation: } {\partial L\over \partial x^{(c)}}={d\over dl}{\partial L\over \partial \dot{x}^{(c)} } \\ & \text{LHS} = {1\over 2}L^{-1}( -{\partial g_{ab}\over \partial x^{(c)}}\dot{x}^{(a)}\dot{x}^{(b)} ) = {1\over 2}( -{\partial g_{ab}\over \partial x^{(c)}}{dx^{(a)} \over d\tau}\dot{x}^{(b)} ) \\ & \text{RHS} ={d\over dl}[ {1\over 2}L^{-1}( -g_{ab}\delta_{ac} \dot{x}^{(b)} -g_{ab}\dot{x}^{(a)}\delta_{bc}) ] ={d\over dl}[ -g_{bc} {d{x}^{(b)} \over d\tau}] \\ & {\text{LHS} \over L} ={\text{LHS} \over L} \\ & \Rightarrow \boxed{ {1\over 2}( -{\partial g_{ab}\over \partial x^{(c)}}{dx^{(a)} \over d\tau}{dx^{(b)} \over d\tau} ) = {d\over d\tau}[ -g_{bc} {dx^{(b)} \over d\tau}] \;\;\;\;\text{Geodesic Equation}}\\ & \Rightarrow \boxed{ 0 = {d^2x^{(b)} \over d\tau^2} \;\;\;\;\text{Geodesic Equation (SR)}} \\ \end{align*}\end{split}\]