# Physics 616

• Prof. Andrew W. Steiner
(or Andrew or "Dr. Steiner")
• Office hour: 103 South College, Thursday 11am
• Email: awsteiner@utk.edu
• Homework: Electronically as .pdf
• You may work with each other on the homework, but you must write the solution in your own words

Use down and up arrows to proceed to the next or previous slide.

## Outline

• Ch. 7 in Schutz

## Equivalence Principles

• Weak Equivalence Principle: Freely falling particles move on timelike geodesics of the spacetime.
• Einstein Equivalence Principle: Any local physical experiment not involving gravity will have the same result if performed in a freely falling inertial frame as if it were performed in the flat spacetime of special relativity.
• Comma $\rightarrow$ semicolon
• Number conservation example $$\left( n U^{\alpha} \right)_{,\alpha} = 0 \quad \Rightarrow \quad \left( n U^{\alpha} \right)_{;\alpha} = 0$$

## Other Conservation "Laws"

• Schutz refers to $$U^{\alpha} S_{,\alpha} = 0$$ as the "law of conservation of entropy"
• Not a law, and $S$ is entropy per particle
• No semicolon necessary. Why?
• We've already seen the conservation of energy and momentum $${T^{\mu \nu}}_{; \nu} = {T^{\mu \nu}}_{; \mu} = 0$$

## Physics in Curved Spacetimes

• Nearly-flat metric $$ds^2 = - \left( 1 +2 \phi\right)~dt^2 + \left( 1- 2 \phi\right) \left( dx^2+dy^2+dz^2\right)$$ with $|\phi|\ll 1$.
• Consider the motion of a freely falling particle (falls on a geodesic) with a nonrelativistic velocity
• For massive particles, given a four-velocity $\vec{U} = d \vec{x}/d \tau$, the four-momentum is $m \vec{U}$
• The geodesic equation $$\nabla_{\vec{U}} \vec{U} =\nabla_{\vec{p}} \vec{p} =0$$ since $\tau/m$ is an affine parameter

## Physics in Curved Spacetimes II

• The geodesic equation is $$\frac{d}{d \lambda} \left( \frac{d x^{\alpha}}{d \lambda} \right) + {\Gamma^{\alpha}}_{\mu \beta} \frac{d x^{\mu}}{d \lambda} \frac{d x^{\beta}}{d \lambda} = 0$$
• Consider the 0 component $$m \frac{d}{d \tau} p^{0} + {\Gamma^{0}}_{\alpha \beta} p^{\alpha} p^{\beta} = 0$$
• And since nonrelativistic, $p^{0}$ is the only significant component (why?), thus $$m \frac{d}{d \tau} p^{0} + {\Gamma^{0}}_{0 0} p^{0} p^{0} = 0$$ with $${\Gamma^{0}}_{00} = \frac{1}{2} g^{0 \alpha} \left( g_{\alpha 0,0} + g_{0 \alpha,0} - g_{0 0,\alpha} \right) = \frac{1}{2} g^{0 0} g_{0 0, 0}$$

## Physics in Curved Spacetimes III

• Continuing $${\Gamma^{0}}_{00} = \frac{1}{2} g^{0 0} g_{0 0, 0} = \frac{1}{2} \left(1 + 2 \phi\right)^{-1} \left(2 \phi\right)_{,0}$$
• We can also replace $(p^0)^2$ with $m^2$ and we can replace $dt$ with $d \tau$ (why?) to get $$\frac{d}{d \tau} p^{0} = -m \frac{\partial \phi}{\partial \tau}$$
• Or the time derivative of the kinetic energy is equal to the additive inverse of the time derivative of the potential energy
• The quantity $p^{0}$ is conserved unless the potential energy is time-dependent
• But $p^{0}$ can be identified with the "energy" only in this frame

## Physics in Curved Spacetimes IV

• Now consider the spatial components of the geodesic equation $$p^{\alpha} {p^{i}}_{,\alpha} = {\Gamma^{i}}_{\alpha \beta} p^{\alpha} p^{\beta} = 0$$
• Again proceeding to lowest order in velocity $$m \frac{d p^{i}}{d \tau} + {\Gamma^{i}}_{00} \left( p^{0} \right)^2 = 0$$
• Again setting $(p^{0})^2 = m^2$ $$\frac{d p^{i}}{d \tau} = - m {\Gamma^{i}}_{00} = 0$$

## Physics in Curved Spacetimes V

• Now evaluating the Christoffel symbol $${\Gamma^i}_{0 0} = \frac{1}{2} g^{i \alpha} \left( g_{\alpha 0,0} + g_{\alpha 0,0} - g_{0 0,\alpha} \right)$$ where $$g^{i \alpha} = \left( 1 - 2 \phi \right)^{-1} \delta^{i \alpha}$$ and thus $${\Gamma^{i}}_{0 0} = \frac{1}{2} \left( 1 - 2 \phi \right)^{-1} \delta^{i j} \left( 2 g_{j 0,0} - g_{0 0, j} \right) = \frac{1}{2} \left( 2 \phi \right)_{, j} \delta^{ij}$$ (missing comma?)
• Putting this all together $$\frac{d p^{i}}{d \tau} = -m \phi_{,i}$$ thus the force is the gradient of the potential

## Energy Conservation

• There is a conservation law associated with $T^{\mu \nu}$, but $p^{0}$ is not, in general, conserved
• There is no single quantity which we can replace $p^{0}$ with that is conserved
• Consider a weak interaction (e.g. beta decay) process inside a neutron star. Does GR modify energy and momentum conservation there?

## Symmetries in the metric

• The geodesic equation implies $$m \frac{d p_{\beta}}{d \tau} = \frac{1}{2} g_{\nu \alpha,\beta} p^{\nu} p^{\alpha}$$
• If the metric (that is, all of it's components) is independent of a coordinate, then the momentum along that direction is constant (i.e. is no force and energy, $p^{0}$, is conserved along that direction)
• In our nearly flat metric from before $$- p_0 \approx m + m \phi + \mathbf{p}^2/(2m)$$
• If we have rotational symmetry, then $p_{\psi} = m r^2 \Omega$ is conserved

## Sign conventions

• One can arbitrarily choose the sign of the metric, the Ricci tensor, and the Einstein tensor.

## Group Work

• Complete 7.5 in Schutz