Relativity

Spacetime diagram take time as one of its axis in its graph. By convention, the positive time is set to be going up.

A point object follows a line through spacetime diagram, called worldline.

The generating set of the Lorentz group contains six elements corresponding to each generator:

\begin{alignat*}{3} B_x(v_x) &= \begin{bmatrix} \gamma_x & -\gamma_x\beta_x & 0 & 0 \\ -\gamma_x\beta_x & \gamma_x & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0& 0 & 0 & 1 \\ \end{bmatrix}, \quad &B_y(v_y) &= \begin{bmatrix} \gamma_y & 0 & -\gamma_y\beta_y & 0 \\ 0 & 1 & 0 & 0 \\ -\gamma_y\beta_y & 0 & \gamma_y & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}, \quad &B_z(v_z) &= \begin{bmatrix} \gamma_z & 0 & 0 & -\gamma_z\beta_z \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -\gamma_z\beta_z & 0 & 0 & \gamma_z \\ \end{bmatrix}, \\ R_x(\theta_x) &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos\theta_x & \sin\theta_x \\ 0& 0 & -\sin\theta_x & \cos\theta_x \\ \end{bmatrix}, \quad &R_y(\theta_y) &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta_y & 0 & -\sin\theta_y \\ 0 & 0 & 1 & 0 \\ 0 & \sin\theta_y & 0 & \cos\theta_y \\ \end{bmatrix}, \quad &R_z(\theta_z) &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta_z & \sin\theta_z & 0 \\ 0 & -\sin\theta_z & \cos\theta_z & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}. \end{alignat*}

where \[ \gamma = \frac{1}{\sqrt{1 - \beta^2}}, \quad \beta = \frac{v}{c}. \]

The Lorentz boost by arbitrary velocity \(\vb{v}\) can be represented by \[ B(\vb{v}) = \begin{bmatrix} \gamma & -\gamma \beta_x & -\gamma \beta_y & -\gamma \beta_z \\ -\gamma \beta_x & 1 + \frac{\gamma^{2}}{1+\gamma}\beta_x^2 & \frac{\gamma^{2}}{1+\gamma}\beta_x\beta_y & \frac{\gamma^{2}}{1+\gamma}\beta_x\beta_z \\ -\gamma \beta_y & \frac{\gamma^{2}}{1+\gamma}\beta_x\beta_y & 1 + \frac{\gamma^{2}}{1+\gamma}\beta_y^2 & \frac{\gamma^{2}}{1+\gamma}\beta_y\beta_z \\ -\gamma \beta_z & \frac{\gamma^{2}}{1+\gamma}\beta_x\beta_z & \frac{\gamma^{2}}{1+\gamma}\beta_y\beta_z & 1 + \frac{\gamma^{2}}{1+\gamma}\beta_z^2 \end{bmatrix}. \]

Let us take the following postulates:

• Laws of physics are the same in all inertial reference frames
• Speed of light in a vacuum \(c\) is the same in all inertial reference frames

In order to measure the light to be the same speed in a new frame, the transformation matrix must be symmetric.

• the unit length of time and the unit length of space of the new frame must be measured to be equal from any frame of reference
• and the angles they make against the path of light must be equal.

First we can match the angle of new time axis based on the relative velocity, and consequently that of new space axis.

\begin{align*} c\tilde{t} &\propto ct - \beta x, \\ \tilde{x} &\propto -\beta ct + x \\ \end{align*}

where \(\beta = v/c\).

There remains one proportionality constant \(\gamma\) that scales both axes at the same time (remember symmetric matrix). It can now be determined by imposing that composition with the reverse transformation has to be equal to the identity transformation.

\begin{align*} \begin{bmatrix} \gamma & -\gamma\beta \\ - \gamma\beta & \gamma \\ \end{bmatrix} &\begin{bmatrix} \gamma & \gamma \beta \\ \gamma \beta & \gamma \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \\ \implies &\gamma^2(1 - \beta^2) = 1 \\ \implies &\gamma = \frac{1}{\sqrt{1 - \beta^2}} \end{align*}

• \(\det \Lambda = 1\).

Time Dilation

1. The time component of the moving frame's time basis vector is measured larger in stationary frame. That means moving clock ticks slower.

Length Contraction

1. The space component of the moving frame's space basis vector is measured larger in stationary frame. That means moving observer experience less distance.
2. The length of a moving object measured by a stationary observer is shorter than the object's proper length, because the observer is looking at different time slice than where the proper length is measured.

Composition of Lorentz matrices are also Lorentz matrix, in which the velocity addition rule can be discovered to be \[ \beta_{u\circ v} = \frac{\beta_u + \beta_v}{1+ \beta_u\beta_v}. \]

Lorentz transformation can also be represented in terms of hyperbolic functions: \[ \Lambda = \begin{bmatrix} \cosh \zeta & -\sinh \zeta \\ -\sinh \zeta & \cosh \zeta \end{bmatrix}. \]

Rapidity is the hypobolic angle \( \zeta \) for this Lorentz boost.

Rapidity satisfies: \[ \zeta = \ln [ \gamma(1+\beta) ] = - \ln [\gamma (1-\beta) ], \] and \[ \zeta = \operatorname{arctanh} \frac{|\mathbf{p}|c}{E} = \frac{1}{2} \ln \frac{E + |\mathbf{p}|c}{E - |\mathbf{p}|c} \]

The set of Lorentz transformations is called the restricted Lorentz group or special indefinite orthogonal group \(\mathrm{SO}^+(1,3)\) (\(+\) for orthochronous), and it is generated by six generators \(K_x, K_y, K_z, J_x, J_y, J_z\) corresponding to three Lorentz boosts \(B_x(\zeta), B_y(\zeta), B_z(\zeta)\) and three rotations \(R_x(\theta), R_y(\theta), R_z(\theta)\).

Explicit matrices of the generators are:

\begin{alignat*}{3} K_x &= \begin{bmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0& 0 & 0 & 0 \\ \end{bmatrix}, \quad &K_y &= \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}, \quad &K_z &= \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{bmatrix}, \\ J_x &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0& 0 & 1 & 0 \\ \end{bmatrix}, \quad &J_y &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ \end{bmatrix}, \quad &J_z &= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}. \end{alignat*}

They generate boost, rotation, and arbitrary Lorentz transformation:

\begin{align*} B(\vb*{\zeta}) &= e^{-\vb*{\zeta}\vdot \vb{K}}, \\ R(\vb*{\theta}) &= e^{-\vb*{\theta}\vdot \vb{J}}, \\ \Lambda(\vb*{\zeta}, \vb*{\theta}) &= e^{-\vb*{\zeta}\vdot\vb{K} - \vb*{\theta}\vdot \vb{J}}. \end{align*}

The boost transforms current coordinates into the coordinates of a new frame with rapidity \(\vb*{\zeta}\) in the current frame, the rotation transforms current coordinates into the coordinates of a new frame rotated by \(\vb*{\theta}\).

The generators are elements of Lie algebra of Lorentz group, and they transfrom as an antisymmetric rank-2 Lorentz tensor. \[ J^{\mu\nu} := \begin{bmatrix} 0 & -K_1 & -K_2 & -K_3 \\ K_1 & 0 & -J_3 & J_2 \\ K_2 & J_3 & 0 & -J_1 \\ K_3 & -J_2 & J_1 & 0 \end{bmatrix} \]

Minkowski metric tensor measures the spacetime interval \(s\) \[ s^2 = \eta_{\mu\nu}x^{\mu}x^{\nu}. \]

Spacetime interval is invariant under Lorentz transformation. Furthermore, Minkowski metric tensor itself is Lorentz invariant. \[ \eta_{\mu\nu} = \Lambda_{\mu}{}^{\rho}\Lambda_{\nu}{}^{\sigma}\eta_{\rho\sigma} . \]

• \(\tau\)
• Eigenzeit

The time measured by stationary frame with respect to the region of interest.

\[ c\tau := \int \dd{s} \] where \(\dd{s}\) is the infinitesimal Spacetime Interval.

\[ u^{\mu} := \dv{x^{\mu}}{\tau} \] where \( \tau \) is the proper time along the trajectory.

Due to the Lorentz invariance of the proper time, 4-velocity is also a Lorentz invariant vector. (The compoenets are covarient, and basis vectors are contravariant.)

In terms of the observer's time \( t \), the observed 4-velocity \( v^{\mu} \) would be: \[ v^{\mu} = \left(1, \dv{x^{i}}{t}\right). \] and the 4-velocity is given as: \[ u^{\mu} = \dv{x^{\mu}}{\tau} = \dv{x^{\mu}}{t} \dv{t}{\tau} = \gamma v^{\mu}. \]

\[ p^{\mu} := mu^{\mu} \] where \( m \) is the rest mass.

The time component represent the energy \[ p^0 = E, \] and the magnitude is given by the rest mass \[ p^{\mu}p_{\mu} = m^2. \]

This can be shown by expanding into Taylor series: \[ cp^0 = \gamma m c^2 = mc^2 + \frac{1}{2} m u^2 + O(\beta^4), \] which contains both the rest energy and Newtonian kinetic energy.

\[ a^{\mu} := \dv{u^{\mu}}{\tau} \]

\[ f^{\mu} := \dv{p^{\mu}}{\tau} \]

The time component is in the unit of power.

Given a spacetime field \( \phi \), the change of \( \phi \) as observed by an observer moving with velocity \( u^{\mu} \): \[ \dv{\phi}{\tau} = u^{\mu} \pdv{\phi}{x^{\mu}}. \]

The derivative is a scalar, and therefore the gradient \( \partial_{\mu}\phi \) is 1-form that maps vector to scalar.

\[ n^{\mu} := n_0 u^{\mu} \] where \( n_0 \) is the rest number density, and \(u^{\mu}\) is the 4-velocity of the particles.

When observed from a moving frame, the density increases by the factor of \( \gamma \) and the flux is given by \( \gamma n_{0}v^i \). This is exactly the Lorentz transformation.

• 4-momentum flux.
• Stress-Energy Tensor, Stress-Energy-Momentum Tensor

\( T^{\mu\nu} \) is the flux of \( \mu \) component of the 4-momentum along the \( \nu \) direction.

For a dust cloud with number flux \(n^{\nu}\), \[ T^{\mu\nu} = p^{\mu}n^{\nu} \] When boosted the energy density \( T^{00} \) increases by the factor of \( \gamma^2 \), once due to the boost in \( p^0 \), and once due to the boost in the number flux \(n^{\nu}\).

\( \epsilon_{ijk} \) gives the volume associated with three vectors.

It is the Doppler effect of the moving source in combination with the time dialation: \[ \nu' = \nu \frac{c}{c-v} \frac{1}{\gamma} = \nu\sqrt{\frac{1+\beta}{1-\beta}}. \]

When the source and receiver are moving tranversely to each other, the time dialation and the longitudinal Doppler effect work in tandom. The exact moment that the longitudinal effect disappears depends on the frame of reference.

The moment at which the source and the receiver is at their closest, the observer sees a blueshifted light \( \gamma \nu \), while at the moment the observer sees the light from the closest point he sees a redshifted light \( \nu/\gamma \).

This is the coordinate system of the frame that is accelerating with proper acceleration \( \alpha \), from the initial position \(c^2/\alpha\) with zero initial velocity. The spacetime is still flat here, so this non-inertial frame may be included in the realm of special relativity.

The coodinate transformation from constant-acceleration frame into the inertial frame is given by:

\begin{align*} ct &= \tilde{x}\sinh(\frac{\alpha}{c^2}c\tilde{t}) \\ x &= \tilde{x}\cosh(\frac{\alpha}{c^2}c\tilde{t}) \\ \end{align*}

where \( \tilde{x} \) and \( \tilde{t} \) are the measured length and proper time of the accelerating frame. The magnitude of proper acceleration can be measured relative to the instantaneous inertial frame tangent to each point on the worldline, and it can be kept constant in those series of frames.

Here, \(\alpha \tilde{t} / c\) can be interpreted as the rapidity of the observer at that time.

The transformation from the inertial frame into the Rindler coordinate is given by:

\begin{align*} \tilde{x}^2 &= x^2 - (ct)^2 \\ c\tilde{t} &= \frac{c^2}{\alpha}\tanh^{-1}\left( \frac{ct}{x} \right) \\ \end{align*}

\(D := c^2/\alpha\) is a convenient unit for Rindler coordinates, sometimes called the Rindler distance.

From the properties of 4-velocity and 4-acceleration,

\begin{align*} u^{\mu}u_{\mu} = c^2 \land u^{\mu}a_{\mu} = 0 \land a^{\mu}a_\mu = \alpha^2 \\ \implies a^t = \frac{\alpha}{c}u^x \land a^x = \frac{\alpha}{c}u^t. \end{align*}

Now we can solve the coupled differential equation in inertial frame:

\begin{cases} \displaystyle \dv{u^t}{\tau} = \frac{\alpha}{c}u^x, \\[0.5em] \displaystyle \dv{u^x}{\tau} = \frac{\alpha}{c}u^t. \end{cases}

The solution with initial condition \(u^x(0) = 0\) is \[ \begin{cases} \displaystyle u^t = c \cosh \left( \frac{\alpha}{c}\tau \right) \\[0.5em] \displaystyle u^x = c \sinh \left( \frac{\alpha}{c}\tau \right) \end{cases}, \begin{cases} \displaystyle a^t = \alpha \sinh \left( \frac{\alpha}{c}\tau \right) \\[0.5em] \displaystyle a^x = \alpha \cosh \left( \frac{\alpha}{c}\tau \right) \end{cases}, \] from which we can calculate the coordinates at time \(\tau\): \[ \begin{cases} \displaystyle ct = \frac{c^2}{\alpha} \sinh(\frac{\alpha}{c}\tau) + t_0, \\[0.5em] \displaystyle x = \frac{c^2}{\alpha} \cosh(\frac{\alpha}{c}\tau) - \frac{c^2}{\alpha} + x_0. \end{cases} \]

If we take the natural initial condition \(t_0 = 0, x_0 = c^2/\alpha\), the spacetime interval from the origin is also constant: \[ s^2 = - \left( \frac{c^2}{\alpha} \right)^2. \] We take the proper time of this reference worldline to be the time coordinate \(\tilde{t}\), and the measured length in this accelerating frame to be \(\tilde{x}\).

Proper Time The points that are constant distance away from the accelrating observer, also form a hyperbola by the argument of similarity, considering that the 4-position is exactly in the direction of acceleration and the space basis vector.

The proper time at a point along the worldline of constant \( \tilde{x} \), differs from the coordinate time \(\tilde{t}\) since it accelerates slower:

\begin{equation*} \tau_{\tilde{x}}(\tilde{t}) = \frac{\tilde{x}}{D}\tilde{t} =: k\tilde{t}. \end{equation*}

The hyperbola of constant \(\tilde{x}\) is scaled version of the reference hyperbola with distance \(D\) from the origin, with respect to the origin by the factor \(k\), \[ \begin{cases} \displaystyle ct = kD \sinh \left( \frac{c\tilde{t}}{D} \right), \\[0.5em] \displaystyle x = kD\cosh \left( \frac{c\tilde{t}}{D} \right) \end{cases}. \] By matching the form to a accelerating worldline, we can see that this is another constant-accelration worldline with its proper time scaled by the same factor \(k\): \[ \begin{cases} \displaystyle ct = kD \sinh \left( \frac{c(k\tilde{t})}{kD} \right) = kD \sinh \left( \frac{c\tau_{\tilde{x}}}{kD} \right), \\[0.5em] \displaystyle x = kD\cosh \left( \frac{c(k\tilde{t})}{kD} \right) = kD \cosh \left( \frac{c\tau_{\tilde{x}}}{kD} \right) \end{cases}. \]

Speed of light The worldline of light that starts at \(\tilde{x}_0\) when \(\tilde{t} = 0\), in the positive \(\tilde{x}\) direction is given by: \[ \tilde{x}(\tilde{t}) = \tilde{x}_0 e^{c\tilde{t}/D}. \] Consequently, the speed of light at \(\tilde{x}_0\) when \(\tilde{t} = 0\) is \[ \dv{\tilde{x}}{\tilde{t}} = \frac{\tilde{x}_0}{D}c. \]

This means that the speed of light is measured to be faster in the direction of the acceleration. This aligns with the proper time being longer further into the direction of acceleration. In one unit of time of the Rindler coordinates, the light move further than usual due to longer proper time.

This does not violate the postulate of special relativity, since this speed of light is measured outside the local inertial frame.

Given two spaceships one after the other that moves forward with the same acceleration, would the string that connected two spaceships in the beginning break after a while?

The answer is yes. The space between them grow bigger as they accelerate and the string breaks.

One of the twin ride a rocket and travel into the deep sea close to the speed of light and come back, while the other remain on the Earth. Who would be older when the rocket come back to the Earth?

The one who took the rocket is older. From the rocket's frame of reference:

1. The proper time goes just a little faster for the rocket when it accelerates to depart the Earth.
2. The time of the other one seems slower for both during the constant velocity travel.
3. The proper time of the Earth goes extremely fast as the rocket turns around, due to the large distance between them.
4. The time of the other one seems slower for both during the constant velocity travel.
5. The proper time goes just a little faster for the rocket when it decelerate to arrive the Earth.

From the Earth's frame of reference:

1. The proper time of the rocket is faster than that of the Earth during the travel.

Over small region of spacetime, the motion of freely falling particles due to gravity cannot be distinguished from uniform acceleration.

In sufficiently small regions of spacetime, we can find a representation such that the laws of physics reduce to those of special relativity.

Gravity falls in a gravitational field in a way indistinguishable from mass.

We try to promote the gravitaional Poisson equation we saw in Newton-Cartan theory into a tensorial equation.

We need to impose divergence free condition on the Ricci tensor making it into Einstein tensor, and find the proportionality constant that matches the Newtonian gravity at low velocity and under weak gravity with time-independent metric.

Einstein field equation can also be derived from Einstein-Hilbert action \[ S = \int \dd[4]x \sqrt{-g}R \] using the principle of stationary action.

The three dimensional embedding of the two dimensional geometry given by Schwarzschild metric for \(r, \phi\) plane. \[ w(r) = 2 \sqrt{r_{\mathrm{s}} (r - r_{\mathrm{s}})} \]

The singularity at \(r = r_{\mathrm{s}}\) is coordinates singularity caused by our choice of coordinates.

The singularity at \(r=0\) is true singularity where theory of general relativity breaks down.

• Einstein Shift

Using the wave covector field \(k_{\mu}\) that are oriented orthogonal to the light beams and exact (non rotating). we can apply it to a normalized time basis vector, and get the frequency/c.

In Schwarzschid coordinates, the frequency \(f\) at \(r\) measured by an observer at infinity is given relative to the frequency \(f_0\) measured when the light reaches the infinity: \[ f = \frac{f_0c}{\displaystyle\sqrt{1 - \frac{r_\mathrm{s}}{r}}} \]

• Wormhole

The singularity at the center of Kruskal-Szekeres coordinate system might be the place where the two "parallel" universe meet.