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Gravitational Physics

Here we lay out Newton's law of gravity and crack open the universe of consequences that spring from it.

Gravitational Redshift

                   

Like the problem of celestial motion, the physical nature of light perplexed the early natural philosophers, including Newton.

Classical electromagnetism—a theory pieced together in the 19th century that unified electric and magnetic forces—revealed light as fast-moving waves on a universe-filling electromagnetic field. In particular, light is not a type of matter; rather, it is energy carried by the electromagnetic field.

This revelation resolved a centuries-long scientific debate and spawned the invention of modern communications technology. However, in an epic series of disruptions early in the 20th century, Einstein called into question our understanding of light, demonstrating that it is measurably affected by gravitational fields.

Here, we are going to explore Einstein's derivation of gravitational redshift, one was the major motivations for Einstein's equivalence principle—the focus of the next quiz.

Einstein began by imagining a particle on a trajectory through Earth's gravitational field.

Suppose a matter particle, mass \(m,\) is a distance \(R\) from the Earth's center. What is the minimum kinetic energy it needs to re-emerge (hint: or escape) from the gravitational pull of the Earth?

Details & Assumptions

  • The mass of the Earth is \(M_E.\)
  • The particle does not interact appreciably with any other matter during its escape.

Suppose the particle began its journey very far from the Earth. How much kinetic energy has it gained when it reaches its point of closest approach, a distance \(R\) from the Earth's center?

Details & Assumptions

  • Like the force of gravity on Earth's surface, Newton's gravitational force is conservative.
  • The mass of the Earth is \(M_E.\)
  • The particle does not interact appreciably with any other matter during its approach.

Under the assumption that the gravitational force is conservative, we can see that a particle passing through the Earth's gravitational field gains some kinetic energy before it losing it again as it escapes.

Here's Einstein's change-up: imagine that the particle in Earth's gravitational field decays, instantaneously losing its mass \(m\) and energizing the local electromagnetic field.

This process is allowed by Einstein's most famous discovery, the mass-energy relation, \[E=mc^2,\] which governs interactions where mass and energy can be exchanged. Because the conversion factor is \(c^2\) is large \((c=\SI[per-mode=symbol]{3.0\times10^8}{\meter\per\second})\), only high-energy processes tend to lose or gain mass.

Einstein understood that this extra energy must be accounted for whenever we apply energy conservation.

Particle decays produce photons, bits of energy we perceive as light. Photons carry the electromagnetic field's energy. This energy is directly related to the frequency \(f\) of the light by \[E=hf\] where \(h=\SI{6.626\times10^{-34}}{\joule\second},\) Planck's constant.


Suppose the particle's decay mints a new photon. The photon then escapes the gravitational field of Earth and we measure its frequency, \(f_\infty\).

If the particle, mass \(m,\) had essentially zero energy before it drifted into the Earth's gravitational field, what is \(f_\infty?\)

Immediately after the particle decay at radius \(R,\) the photon's frequency is \(f_R.\)

How does \(f_R\) compare to measured frequency \(f_\infty\)?

The redshift factor \(z\) is defined as the photon's frequency shift relative to the measured frequency \(f_\infty\): \[z=\frac{\Delta f}{f_\infty}=\frac{f_R - f_\infty}{f_\infty}.\]

Derive an expression for the redshift factor \(z\) as the photon passes out of the Earth's gravitational field.

We have replicated an argument developed by Einstein that shows, remarkably, that a gravitational field shifts the frequency of light.

We demanded that energy be conserved as a particle decays to a photon while moving through a gravitational field, and we were led to a prediction of gravitational redshift. Specifically, the frequency of a photon decreases (it shifts toward red) as it leaves a region of stronger gravitational field.

In general, we can express the result in terms of a change in gravitational potential energy \(\Delta U_\text{grav}\) of a mass \(m\): \[\frac{\Delta f}{f_\infty} = \frac{\Delta U_\text{grav}}{mc^2}\] Note that \(m\), the mass of a hypothetical particle, will cancel between the numerator and denominator.

We assumed that the photon carries all of the particle's energy, but Einstein's argument also applies when the escaped photon's energy is a fraction of the total. A common particle decay is a neutral pion (\(\pi^0\)) into two photons. You might check that as long as the energy of the particle is conserved between the two photons, you arrive at the same expression for redshift factor \(z.\)

Is gravitational redshift significant in communications between Earth's surface and orbit, or even in ground-based communications?


The International Space Station (ISS) has receivers in the \(K_u\) frequency band near \(f=\SI{15}{\giga\hertz}.\) What is the order of magnitude of the frequency shift (in \(\si{\hertz}\)) attributable to gravitational redshift?

Details & Assumptions

  • The orbit of the ISS is \(\SI{400}{\kilo\meter}\) above Earth's surface.
  • Treat the gravitational field between the ISS and Earth's surface as uniform.

Gravitational redshift was measured directly in 1960 in a test at Harvard University. Remarkably, the vertical height between emitter and receiver was only \(\SI{22.5}{\meter}.\) The experimenters used a source of high-frequency gamma rays (an unstable iron isotope) instead of visible light to detect the frequency shift of \(\sim \SI{50}{\hertz}.\)

Newton understood gravity as a force between pairs of masses. Einstein's insights of the early 20th century turned this understanding on its ears, demonstrating that energy conservation requires an interaction between massless light and gravity. Surprisingly, gravitational redshift is no small effect; some signals between Earth's surface and orbiting spacecraft have to be corrected for the frequency shift.

In the next quiz, we will examine gravitational redshift through the lens of the equivalence principle to try to make sense of why light is influenced by the gravitational field.

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