What's the biggest black hole that could pass through the inner solar system and not kill everyone

Factual Questions
81

To add some words to the points just made…

If you are in your spaceship in free space accelerating at 9.8 m/s[sup]2[/sup], your accelerometer would indicate that. You could use a handheld accelerometer, or you could use the one you always have with you: your inner ears’ semicircular canals, which give you your sense of balance and acceleration.

If your spaceship had no windows, you could say nothing more than “I’m accelerating forward, it seems.” If your spaceship were instead sitting stationary on the surface of earth, pointed upward, and it still had no windows, your accelerometer(s) would say the same thing: 9.8 m/s[sup]2[/sup]. If someone lied to you to make you think you really were in free space, you would simply say “I’m accelerating forward, it seems.”

In everyday language, you probably would choose to say that you are stationary in the second (Earth surface) scenario. But there’s nothing you can do to distinguish these two cases. If you are willing to call one “stationary”, then the other could also be called “stationary”.

If you want to leave gravity out of it, then it’s just a matter of care with the words. Say you are walking along a sidewalk and you pass a mailbox and a mailman, and the mailman is walking in the other direction. You could ask:

  • Where am I relative to the mailbox? (Answer: 0 feet away, then a short while later 10 feet away, then a short while later 20 feet away…)
  • Where am I relative to the mailman? (Answer: 0 feet away, then a short while later 20 feet away, then a short while later 40 feet away…)
  • Where am I relative to me? (Answer 0 feet away, then a short while later 0 feet away, then a short while later 0 feet away…)

If you were accelerating at a constant rate relative to the sidewalk and were interested in your velocity, you could ask:

  • What is my velocity relative to the mailbox? (5 ft/s, then 10 ft/s, then 15 ft/s)
  • What is my velocity relative to the mailman? (10 ft/s, then 15 ft/s, then 20 ft/s)
  • What is my velocity relative to me? (0 ft/s, then 0 ft/s, then 0 ft/s)

All this is saying is that one must be explicit about which velocity you’re interested in.

82

I’m not sure your math works on the acceleration example.

But the principle is the same as the velocity example.

manson1972, what I am talking about is the basics of relative motion. It works in tandem with the vector mathematics that form the principles of linear motion, and eventually dynamics.

In order to define a distance, you have to define a distance from some point. That is the reference point. Say you want to specify a location in your living room. You can, for example, tell how many feet it is from the door, but that only defines one direction. Your room is (presumably) a box. You could map it with a flat layout showing the floorspace, a projection from above. To specify any location in the room, you need two directions. Typically, that is done by defining an X axis and a Y axis. Where those cross is the “origin”, i.e. the reference point. That point is the location that is (0,0) in the coordinate axes. Then any location in that room is two numbers - the distance from the origin in the X axis and the distance from the origin in the Y axis.

The interesting thing is that if you want to tell how far apart two objects in the room are, you can do so if you know both locations as defined by that same axis system. It’s simple vector math - Subtract the X values, subtract the Y values.

But the catchy thing is the velocities and accelerations are also vectors, and also can be defined with an axis system in just the same way. And the same kinds of math can be done, as long as they have the same zero point, reference point.

But with vector math, you can change reference points. Instead of using the corner nearest the door, you can use the corner nearest the window, or the cable jack on the wall, or the light switch, or any other point as your reference. As long as you keep all the values straight, you can relate the same point in two different reference systems by simple math of the difference between the two reference points.

So when you talk about being on a rocket and measuring an acceleration, but you are using that rocket as your reference point, the acceleration you measure is not being applied to the rocket, it is being applied to the rest of the world. Otherwise, you are changing your reference point. You can do the math that way, but you can’t simultaneously apply the acceleration in one reference frame and your velocity in another reference frame and your position in another reference frame and expect a meaningful answer. You have to keep your reference frames the same (or apply the correction factors, which amounts to the same thing).

Yes, an accelerometer is going to measure your acceleration. But you still have to correspond that with the reference system that you are using to measure your length contraction or time dilation. Putting those in the same system either gives a reference from your starting point, or gives you a velocity of zero.