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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.

Equivalence Principle

                       

While Newton originally formulated gravity as a universal force between any pair of masses, the last quiz demonstrated that gravity's influence extends to massless photons.

Einstein's prediction of gravitational redshift led to speculation that the effects of gravity may have a more basic underlying cause. This intuition is captured in the equivalence principle, which can be stated in two forms:

  • Weak form: inertial and gravitational masses are equal.
  • Strong form: the effects of gravity are indistinguishable from the effects arising in a particular accelerating frame of reference.

In the Explore, we saw that the weak form guarantees that the gravitational field has the same effect on all masses. The strong form embodies Einstein's observation that gravitational effects can be simulated by a non-inertial frame of reference. In this quiz we will discover how the weak and strong forms of the principle are related, and we will take another look at gravitational redshift.

Suppose you are accelerating downward in an elevator at a high rate as you accidentally drop a \(\SI{60}{\gram}\) chicken egg.

For what downward accelerations \(\mathbf{a}\) of the elevator does the egg appear to fall upward?

Details

  • You are close enough to Earth's surface that the gravitational field is about \(\SI[per-mode=symbol]{9.8}{\meter\per\second\squared}.\)

Now, suppose inertial mass of the egg is not equal to the gravitational mass. In fact, \(m_I = 0.9 m_G.\)

What is the minimum downward acceleration \(\mathbf{a}\) the elevator must have so that the \(\SI{60}{\gram}\) chicken egg you accidentally drop appears to fall upward?

Details

  • Express your answer in \(\si[per-mode=symbol]{\meter\per\second\squared}.\)
  • Use \(g=\SI[per-mode=symbol]{9.8}{\meter\per\second\squared}.\)

Non-inertial frames can negate the effect of gravity as in a downward-accelerating elevator; or they can induce a fictitious force that mimics gravity to an non-inertial observer, as in the gravitron ride.

Einstein wondered: could an observer, such as the elevator passenger in the previous problems, perform a test to distinguish between the effect of a gravitational field and a non-inertial frame? Provided inertial and gravitational masses are equal for all materials, an anomaly that no test has ever detected, the answer is no.

Thus, Einstein's equivalence principle asserts that any gravitational effect, including gravitational redshift, can be understood as a non-inertial-frame effect. In particular, the gravitational redshift arising in a uniform field on Earth's surface should be equivalent to redshift of light arising in a uniformly accelerating rocket.

Whether this is true is what we'll explore in the remainder of this quiz.

Let's imagine a large rocket that is accelerating uniformly through a region of space where there is no gravity. A person near the base of the rocket (A) is entertaining their cat in the cone of the rocket (B) with a laser pointer. We would like to

  • understand why the laser frequency detected by the cat would be shifted in a uniformly accelerating frame;
  • calculate the magnitude of the shift \(\Delta f\) to compare to the gravitational redshift we calculated in the previous quiz.


What should the acceleration \(a\) of the rocket be to simulate a uniform gravitational field \(g\) between A and B, pointing toward the left (toward A)?

To understand why the frequency of the laser shifts as it propagates along the length of the rocket, you aim your laser at the cone of the rocket, a distance \(H\) from the base, and turn it on.

The front of the light beam travels with speed of light \(c\) into the cone of the rocket during time \(\Delta t_\text{trav.}.\) However, during this time the cone continues to move as the rocket accelerates at a rate \(g.\)

Use kinematics to derive an equation that can be solved for \(\Delta t_\text{trav.}.\)

How much faster \(\Delta v\) is the rocket moving when the laser light is first seen by the cat?

Because the laser light requires a small but finite time \(\Delta t_\text{trav.}\) to travel to the cat, the cat is moving with a relative speed \(\Delta v\) along the wave's trajectory when it detects the light.

Relative motion of a wave source and wave detector produces a frequency shift known as the Doppler effect. How is \(f_A,\) the frequency of the laser light as it is emitted, related to \(f_B,\) the frequency detected by the cat?

Hint: Frequency is measured as the rate at which the peaks of a wave are passing a detector.

We can work out the relationship between \(f_A\) and \(f_B\) by considering how long it takes one full wavelength of the light to pass the cat.

Suppose the laser pointer emits a single wave during time \(T_A.\) The same wave takes time \(T_B\) to pass the cat.

If the cat is moving at a speed \(\Delta v\) relative to the laser pointer when it emitted the wave, use kinematics to relate \(T_A\) and \(T_B.\)

Details & Assumptions

  • The speed of the light wave is \(c.\)
  • The speed of the rocket increases negligibly as the wave passes the cat.

Now we are ready to re-discover an expression for the red-shifted frequency detected by the cat, \(f_B.\)

Use the previous result to relate \(f_B=1/T_B\) to \(f_A=1/T_B\). Then re-express the relationship in terms of \(g\) and \(H\) by eliminating \(\Delta v\).

What is the resulting frequency relationship?

According to the equivalence principle, the frequency relationship derived from the Doppler effect is physically equivalent to the gravitational redshift Einstein predicted using energy conservation.

However, two different approaches produced two asymptotically equivalent relationships. Which one is correct?

General relativity, a modern theory of gravity developed by Einstein, makes yet another prediction for the redshift in the gravitational field of a spherical mass \(M\): \[f_B = f_A \sqrt{\frac{1-\frac{2GM}{c^2r_A}}{1-\frac{2GM}{c^2r_B}}},\] which shares some of the features of both of our previous predictions.

General relativity has been extensively tested and predicts a variety of phenomena that Newtonian gravity cannot explain. Unfortunately, no analog of a uniform gravitational field is possible in general relativity, but we can set \(r_A=R_E,\) Earth's radius, and \(r_B=R_E+H.\) We can identify \(g = GM_E/R_E^2\) and rewrite \[f_B = f_A \sqrt{\frac{1-\frac{2gR_E}{c^2}}{1-\frac{2gR_E}{c^2(1+H/R_E)}}}.\] Taylor expanding first in small parameter \(H/R_E\) then in \(gR_E/c^2,\) we find \[f_B = f_A \sqrt{1-\frac{2gH}{c^2}}.\] This is identical to the relationship obtained from the equivalence principle.

If we return to our free-falling elevator and think about how gravitational redshift arose from applying the equivlence principle, we can ponder one of the most heralded ideas in physics.

As the elevator falls freely through a gravitational field, all gravitational effects vanish—even gravitational redshift. The acceleration of the elevator negates the gravitational field for the observers inside.

This insight led Einstein to speculate that the action of gravity (free-fall) might arise from some very basic property of space (and time) itself. In his theory of general relativity, he shows conclusively that this basic property is the spacetime's intrinsic curvature, a consequence of the presence of nearby mass. Einstein's equation relates distributions of mass and energy to spacetime curvature.

We will return to this far-reaching idea in the final chapter, Newtonian Cosmology, of this course. For now, we will continue to think of gravity as a vector field set up in space by nearby masses. In the next chapter, we will work out more sophisticated ways of calculating gravitational field before we move on to study Keplerian orbits.

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