Problem 40 - Calculus

Question

One of Einstein's most amazing predictions was that light traveling from distant stars would bend around the sun on the way to earth. His calculations involved solving for \(\phi\) in the equation $$ \sin \phi+b\left(1+\cos ^{2} \phi+\cos \phi\right)=0 $$ where \(b\) is a very small positive constant. (a) Explain why the equation could have a solution for \(\phi\) which is near 0 (b) Expand the left-hand side of the equation in Taylor series about \(\phi=0,\) disregarding terms of order \(\phi^{2}\) and higher. Solve for \(\phi .\) (Your answer will involve b.)

Step-by-Step Solution

Answer

(a) Because small \( \phi \) simplifies the equation. (b) \( \phi \approx -3b \).

Step 1: Understanding the Problem

We are given an equation where the objective is to solve for \( \phi \). It's stated that \( b \) is a very small constant. The equation hints that small angles \( \phi \) might be of interest due to the presence of trigonometric functions.

Step 2: Justifying a Solution Near Zero

For small values of \( \phi \), \( \sin \phi \approx \phi \) and \( \cos \phi \approx 1 \). This means for very small \( \phi \), the left hand side of the equation simplifies, likely allowing \( \phi = 0 \) to be a possible solution. The small constant \( b \) allows us to ignore higher-order terms.

Step 3: Expanding in Taylor Series

We expand \( \sin \phi \) and \( \cos \phi \) using a Taylor series around \( \phi = 0 \). For \( \sin \phi \), it is \( \phi \) and for \( \cos \phi \), it is \( 1 - \frac{\phi^2}{2} + \ldots \). Since terms involving \( \phi^2 \) and higher are disregarded, \( \cos \phi \approx 1 \) and \( \cos^2 \phi \approx 1 \).

Step 4: Substituting Taylor Expansions into the Equation

Replacing \( \sin \phi \) with \( \phi \) and both \( \cos \phi \) and \( \cos^2 \phi \) with \( 1 \) in the original equation: \[ \phi + b(1 + 1 + 1) = 0 \] Simplifies to: \[ \phi + 3b = 0 \]

Step 5: Solving for \( \phi \)

From \( \phi + 3b = 0 \), solve for \( \phi \): \[ \phi = -3b \] Since \( b \) is small, \( \phi \) is a small number, thus validating our approximation.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles to the ratios of sides in right triangles. They are fundamental in the study of periodic phenomena and appear often in engineering and physics analyses.

• The most common trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)).
• Sine and cosine functions are especially important in wave motion and oscillations.
• These functions can also be expressed as infinite series, known as Taylor or Maclaurin series, which provide approximations that are crucial in small angle approximations.

In the context of Einstein's prediction, trigonometric functions like sine and cosine help in simplifying and solving equations for angles small enough, seen in bending streams of light.
Understanding how to convert \( \sin \phi \) to \( \phi \) and \( \cos \phi \) to\( 1 \) via their approximations becomes essential for solving near-zero conditions.

Small Angle Approximation

The small angle approximation is a simplification technique used in trigonometry when dealing with angles that are very small, typically measured in radians. This approximation makes it easier to solve equations that involve trigonometric functions.

• For small angles, which are close to zero, \( \sin \phi \) can be approximated to \( \phi \) and \( \cos \phi \) to \( 1 \).
• Similarly, \( \tan \phi \) is also approximated to \( \phi \) for such angles.

These approximations arise from the Taylor series expansion of sine and cosine functions.
By applying the small angle approximation to Einstein's equation, \( \sin \phi + b(1 + \cos^2 \phi + \cos \phi) = 0 \), the complex trigonometric forms reduce to simpler linear forms, paving the way to solve for \( \phi \) without significant computations.
Thus, calculations become more manageable, especially in physics where minute angles often appear, such as the bending of light around large celestial bodies like the sun.

Einstein's Prediction

Einstein's prediction regarding light bending around a massive object, such as the sun, comes from his revolutionary theory of General Relativity. This concept fundamentally changed how we perceive gravity and light.

• Gravity, according to Einstein, is not merely a force but a curvature in spacetime caused by mass.
• Light, while traveling through space, follows the curvature, appearing to bend from the perspective of an outside observer.
• This prediction was famously confirmed during a solar eclipse in 1919, dramatically reshaping modern physics.

The mathematical expressions and concepts, like small angle approximations, are rooted in such predictions.
In the given exercise, Einstein's equation incorporates trigonometrical behavior with constants, showcasing how theoretical predictions can be tangibly linked with mathematical calculations.
It highlights the predictive power of equations, enabling the estimation of light's path in cosmological setups.