# Gravity

## Contents

- 1 Problem 1: Doppler effect with constant acceleration
- 2 Problem 2: on the Equivalence Principle
- 3 Problem 3: gravity and antimatter
- 4 Problem 4: EM and gravity
- 5 Problem 5: gravity in atoms
- 6 Problem 6: gravity and quantization
- 7 Problem 7: gravity scale
- 8 Problem 8: Jeans instability
- 9 Problem 9: star clusters
- 10 Problem 10: the Jeans criterion
- 11 Problem 11: typical critical density
- 12 Problem 12: gravitational pressure
- 13 Problem 13: the dynamic time
- 14 Problem 14: another estimation for dynamic time
- 15 Problem 15: negative thermal capacity
- 16 Problem 16: gravitational and cosmological redshifts
- 17 Problem 17: gravitational redshift
- 18 Problem 18: photons in gravitational field
- 19 Problem 19: light deflection
- 20 Problem 20: local clocks

### Problem 1: Doppler effect with constant acceleration

Consider two observers at constant distance $L$ between them, moving far from any mass, i.e. in the absence of gravitational field with constant acceleration $a$. At $t_0$ the rear observer emits a photon with wavelength $\lambda$. Calculate the redshift that the leading observer will detect.

### Problem 2: on the Equivalence Principle

Let's suppose that an elevator's rope breaks and elevator enters the state of free fall state. Is it possible to determine experimentally, been inside, that the elevator is falling near Earth's surface?

### Problem 3: gravity and antimatter

Richard Feynman wrote: "The striking similarity of electrical and gravitational forces $\ldots$ has made some people conclude that it would be nice if antimatter repelled matter;". What arguments did Feynman use to demonstrate the inconsistency of this assumption (at least in our world)?

### Problem 4: EM and gravity

What is the difference (quantitative and qualitative) between the gravitational waves and the electromagnetic ones?

### Problem 5: gravity in atoms

Find the probability that transition between two atomic states occurs due to gravitation rather than electromagnetic forces.

### Problem 6: gravity and quantization

In his Lectures on Gravitation Feynman asks: "$\ldots$maybe nature is trying to tell us something new here, maybe we should not try to quantize gravity. Is it possible perhaps that we should not insist on a uniformity of nature that would make everything quantized?". And answers this question. Try to reproduce his arguments.

### Problem 7: gravity scale

Evidently the role of gravitation grows with the mass of a body. Show that gravitation dominates if the number of atoms in the body exceeds the critical value \(N_{cr}\simeq(\alpha/\alpha_G)^{3/2}\simeq10^{54},\) where $\alpha=e^2/(\hbar c)$ is the fine structure constant and $\alpha_G\equiv Gm_p^2/(\hbar c)$ is the *fine structure constant* for gravitation, $m_p$ is proton's mass.

### Problem 8: Jeans instability

Stars form from gas and dust due to gravitational instability, which forces gas clouds to compress. This process is known as Jeans instability after the famous English cosmologist James Jeans (1877 -- 1946). What is the physical cause of the Jeans instability?

### Problem 9: star clusters

Observations show that stars form not individually, but in large groups. Young stars are detected in clusters, which contain, usually, several hundreds of stars, which were formed at the same time. Theoretical calculations show, that formation of individual stars is almost impossible. How could this claim be justified?

### Problem 10: the Jeans criterion

A gas cloud of mass $M$ consisting of molecules of mass $\mu$ is unstable if the gravitational energy exceeds the kinetic energy of thermal motion. Derive the stability condition for the spherically symmetric homogeneous cloud of radius $R$ (the Jeans criterion).

### Problem 11: typical critical density

Estimate the critical density for a hydrogen cloud of solar mass at temperature $T=1\ 000\ K$.

### Problem 12: gravitational pressure

Compare the gravitational pressure in the centers of the Sun ($\rho=1.4\ g/cm^3$) and the Earth ($\rho=5.5\ g/cm^3$).

### Problem 13: the dynamic time

Gravitational forces on the Sun are balanced by gas pressure. If pressure *switches of* at some moment, Sun would collapse. The time of gravitational collapse is called the dynamic time. Calculate this time for Sun.

### Problem 14: another estimation for dynamic time

Show that for any star its dynamic time is approximately equal to the ratio of star's radius to the escape velocity on its surface. For the Sun, compare this estimate with the exact value obtained in the previous problem.

### Problem 15: negative thermal capacity

Show that stars (and star clusters) have an amazing property of negative thermal capacity: the more it looses energy due to radiation from its surface, the higher is the temperature in its center.

### Problem 16: gravitational and cosmological redshifts

Along with cosmological redshift, there is the gravitational redshift (the effect of General Relativity), which consists in change in clock under varying gravitational potential. How could these effects be distinguished qualitatively?

The following four problems are based on the paper [Okun L B "The theory of relativity and the Pythagorean theorem" Phys. Usp. 51 622–631 (2008)]. Here we try to understand the effect of gravity on matter (particles and photons) in terms of Newtonian and relativistic physics, but without relying on the General Theory of Relativity (GTR). Thus the terminology differs from that of GTR, and one should not be surprised to see that i.e. $c$ is not constant. For readers familiar with GTR it would be instructive to reformulate the solutions in its terms and establish the relations between the notions used in both approaches.

### Problem 17: gravitational redshift

Demonstrate, that photon emitted at lower floor of a building due to the transition between two nuclear (atomic) levels could not induce the reverse transition in the same nucleus (atom) at higher floor.

### Problem 18: photons in gravitational field

Consider a photon in static gravitational field. Which characteristics of this photon (energy, frequency, momentum, wavelength) change and which remain the same?

### Problem 19: light deflection

Describe the mechanism of light deflection in the gravitational field of the Sun and galaxies.

### Problem 20: local clocks

Trace the photon from problem on gravitational redshift using clocks placed along the photon's trajectory, at each floor.