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There are two prevailing ideas of what this matter could be—WIMPs and MACHOs. WIMPs stands for weakly interacting massive particles. These particles (neutrinos are one example) interact very weakly with ordinary matter and, hence, are very difficult to detect directly. MACHOs stands for massive compact halo objects, which are composed of ordinary baryonic matter, such as neutrons and protons. There are unresolved issues with both of these ideas, and far more research will be needed to solve the mystery.
Newton’s law of gravitation | ${\overrightarrow{F}}_{12}=G\frac{{m}_{1}{m}_{2}}{{r}^{2}}{\widehat{r}}_{12}$ |
Acceleration due to gravity
at the surface of Earth |
$g=G\frac{{M}_{\text{E}}}{{r}_{}^{2}}$ |
Gravitational potential energy beyond Earth | $U=-\frac{G{M}_{\text{E}}m}{r}$ |
Conservation of energy | $\frac{1}{2}m{v}_{1}^{2}-\frac{GMm}{{r}_{1}}=\frac{1}{2}m{v}_{2}^{2}-\frac{GMm}{{r}_{2}}$ |
Escape velocity | ${v}_{\text{esc}}=\sqrt{\frac{2GM}{R}}$ |
Orbital speed | ${v}_{\text{orbit}}=\sqrt{\frac{{\text{GM}}_{\text{E}}}{r}}$ |
Orbital period | ${\rm T}=2\pi \sqrt{\frac{{r}^{3}}{{\text{GM}}_{\text{E}}}}$ |
Energy in circular orbit | $E=K+U=-\frac{Gm{\text{M}}_{\text{E}}}{2r}$ |
Conic sections | $\frac{\alpha}{r}=1+e\text{cos}\theta $ |
Kepler’s third law | ${{\rm T}}^{2}=\frac{4{\pi}^{2}}{GM}{a}^{3}$ |
Schwarzschild radius | ${R}_{\text{S}}=\frac{2GM}{{c}^{2}}$ |
The principle of equivalence states that all experiments done in a lab in a uniform gravitational field cannot be distinguished from those done in a lab that is not in a gravitational field but is uniformly accelerating. For the latter case, consider what happens to a laser beam at some height shot perfectly horizontally to the floor, across the accelerating lab. (View this from a nonaccelerating frame outside the lab.) Relative to the height of the laser, where will the laser beam hit the far wall? What does this say about the effect of a gravitational field on light? Does the fact that light has no mass make any difference to the argument?
The laser beam will hit the far wall at a lower elevation than it left, as the floor is accelerating upward. Relative to the lab, the laser beam “falls.” So we would expect this to happen in a gravitational field. The mass of light, or even an object with mass, is not relevant.
As a person approaches the Schwarzschild radius of a black hole, outside observers see all the processes of that person (their clocks, their heart rate, etc.) slowing down, and coming to a halt as they reach the Schwarzschild radius. (The person falling into the black hole sees their own processes unaffected.) But the speed of light is the same everywhere for all observers. What does this say about space as you approach the black hole?
What is the Schwarzschild radius for the black hole at the center of our galaxy if it has the mass of 4 million solar masses?
$1.19\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{7}\text{km}$
What would be the Schwarzschild radius, in light years, if our Milky Way galaxy of 100 billion stars collapsed into a black hole? Compare this to our distance from the center, about 13,000 light years.
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