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Mobile App Chapter 8: Problem 11
How close must you be (in terms of \(R_{\mathrm{s}}\) ) to a \(3 \mathrm{M}_{\odot}\) black hole to find that a clock runs at \(10 \%\) the rate it runs when it is far away?
Short Answer
Expert verified
To find that a clock runs at 10% of its far-away rate, you must be about 1.01 times the Schwarzschild radius (\r) from a 3M_{\text{\odot}} black hole.
Step by step solution
01
Understand the problem
We need to determine the distance at which a clock runs at 10% of its rate far from the black hole. This involves gravitational time dilation near a black hole.
02
Recall the Schwarzschild Radius
The Schwarzschild Radius (\r) for a black hole is given by \(R_{\text{s}} = \frac{2GM}{c^2}\), where \(G\) is the gravitational constant, \(M\) is the mass of the black hole, and \(c\) is the speed of light. For a black hole of mass \(3M_{\text{\odot}}\), compute \(R_{\text{s}}\).
03
Calculate Schwarzschild Radius
Substitute the values: \(G = 6.674 \times 10^{-11} \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\), \(M = 3 \times 1.989 \times 10^{30} \text{kg}\) (solar mass), and \(c = 3 \times 10^8 \text{m/s}\) into the Schwarzschild radius formula. \[ R_{\text{s}} = \frac{2 \times 6.674 \times 10^{-11} \times 3 \times 1.989 \times 10^{30}}{(3 \times 10^8)^2} \approx 8.86 \times 10^3 \text{m} \]
04
Time Dilation Formula
Use the gravitational time dilation formula: \[ t_f = t_0\sqrt{1 - \frac{R_s}{R}} \]where \(t_f\) is the time far away, \(t_0\) is the affected time, \(R_s\) is the Schwarzschild radius, and \(R\) is the distance from the black hole.
05
Apply given conditions
Given that the clock runs at \(10\text{%}\) of its rate, \( \frac{t_0}{t_f} = 0.10\). Therefore, \[ \left(\frac{t_0}{t_f}\right)^2 = 0.01 = 1 - \frac{R_s}{R} \] Solve for \(R\).
06
Solve for distance \(R\)
Rearrange the equation to solve for \(R\): \[ \frac{R_s}{R} = 1 - 0.01 = 0.99 \] \[ R = \frac{R_s}{0.99} \approx \frac{8.86 \times 10^3 \text{m}}{0.99} \approx 8.95 \times 10^3 \text{m} \]
07
Express in terms of \(R_{\text{s}}\)
Thus, \[ R \approx 1.01R_s \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Schwarzschild radius
The Schwarzschild radius, denoted as \(R_{\text{s}}\), is a critical radius associated with every mass, where the escape speed equals the speed of light. This concept is vital for understanding black holes, as it defines the event horizon or the 'point of no return'. For any object compressed within this radius, it will inevitably collapse into a black hole.
- Formula: \(R_{\text{s}} = \frac{2GM}{c^2}\)
- \(G\): gravitational constant (~\(6.674 \times 10^{-11} \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\))
- \(M\): mass of the object
- \(c\): speed of light (~\(3 \times 10^8 \text{m/s}\))
mass of black hole
The mass of a black hole is a key factor in determining its gravitational influence and the radius of its Schwarzschild radius. Black holes can vary in mass, from a few times more than the mass of our Sun (solar mass, \(M_{\text{odot}}\)) to millions or even billions of solar masses.
- In astronomy, the mass of a black hole is often expressed in terms of solar masses.
- The mass greatly impacts the curvature of spacetime around the black hole, important for understanding gravitational time dilation.
Einstein's theory of relativity
Einstein's theory of relativity, including both special and general relativity, revolutionized our understanding of space, time, and gravity.
- Special Relativity: Addresses the relationship between space and time in the absence of gravitational fields, establishing the speed of light as a constant.
- General Relativity: Extends this to include gravity, describing it as the warping of spacetime by mass and energy.
- Gravitational time dilation - Clocks run slower in stronger gravitational fields.
- Black holes - Regions where spacetime curvature becomes extreme, leading to event horizons.
- Light bending - Massive objects can bend light, an effect known as gravitational lensing.
time dilation near a black hole
Time dilation near a black hole is a direct consequence of Einstein's general theory of relativity. In the presence of a strong gravitational field, time passes more slowly relative to regions with weaker gravitational fields.
- This effect is described mathematically by the gravitational time dilation formula: \[ t_f = t_0\frac{R}{R_s} \]
- \(t_f\): proper time far from the black hole.
- \(t_0\): proper time close to the black hole.
- \(R\): distance from the black hole where the clock is being observed.
- \(R_{\text{s}}\): Schwarzschild radius.
- This highlighted the dramatic effect of time dilation near a black hole, showing that to observe such a slowing, one must be very close to the Schwarzschild radius, precisely around \(1.01R_{\text{s}}\).
