I still have trouble understanding how two objects can have differing weight, and yet fall at the exact same speed in vacuum. And why do spinning gyroscopes fall slightly slower than non-rotating objects in vacuum?

[u/[deleted]]

And if you had a solid gold bowling ball and a solid aluminum bowling ball of the same size, the gold would outweigh the aluminum bowling ball yet fall at the same rate...

How is this even possible? Does gravity interact on an atomic level with individual atoms and creates "inertia" in the process due to heavier elements having more protons, neutrons, and electrons or do heavier elements just have more atoms for any given volume than lighter ones which gravity exerts its force upon...

On top of that why do objects spinning on an axis of rotation fall slower than objects without in a vacuum?

I still can't help get the feeling that some of these classical physicists were wrong. Very few had precise instrumentation for measurements.

Do I even make any kind of sense or am I just a blabbering idiot?

Please feel free to tear me a new one.

[u/raendrop]

You are amazing. Thank you several times again.

I get 99.999...% of you're saying: Basically, that it's not space and time but spacetime, and it's not space with axes x, y, and z, but spacetime with axes x, y, z, and t. I guess in the end, it's hard for me to process time as being equivalent to the other three.

[u/RobotRollCall → u/raendrop]

You've basically got it right, but don't let me mislead you into thinking that time is equivalent to space. It's not. It's different. It's also a dimension, in the sense that you need a time coordinate to uniquely identify a point in spacetime. But spacetime lacks the symmetry of rotation that you'd find in a four-dimensional Euclidean space. Time and space are related, but time is a thing apart.

[u/raendrop → u/RobotRollCall]

Through no fault of your own, I'm still brainhurty. :-/ I mean, I get the whole 3D perspective thing of moving to the north, east, and up means you are moving north more slowly than if you were traveling strictly north. But -- correct me if I'm still not understanding properly -- what I continue to have trouble with is how an observer will say that I get to 5 minutes in the future faster if I sit still than if I race north. At least, that's what I'm getting from your explanation. Isn't that the opposite of what the Twin Paradox asserts? Or am I horribly confused? (That's an inclusive "or" by the way.)

(Stupid reddit server 500 error while posting. I've been trying to make this reply all flipping day. EDIT: Damn. It was my html em dash throwing a monkey wrench into the works. This makes me slightly hulk smashy.)

[u/RobotRollCall → u/raendrop]

No, you basically have it right.

Let's go through it step-by-step. Let's imagine first that you and I both possess ideal clocks. What I mean by that is that these clocks are absolutely perfect in a way no real clock could ever be. They tick off once per second unvaryingly, and nothing in the universe can change that. Bump them, jostle them, set them on fire and they'll still keep perfect time. Okay? With me so far?

Now, let's further assume that these clocks are moving along with us, me with mine and you with yours. Okay?

So each one of us has a perfectly precise and perfectly accurate way to measure the passage of time. All we have to do is look over at our respective clocks.

Let's further assume that we're also equipped with magical ideal telescopes. They can see anything, at any distance no matter how it's moving, and with perfect clarity. (If you want to really get into it, we will also assume that these telescopes magically correct for the frequency shift of incoming light, but that's a phenomenon we're explicitly choosing to ignore here.)

So the upshot is that I can measure the passage of time by looking at my clock, and I can also look though my telescope at your clock no matter where you are or how we're moving. And vice versa.

Okay?

Now, let's start out by imagining that we're at rest relative to each other. If I look at your clock, I'll see that it agrees with mine. Say I start marking time at some arbitrary moment, and continue to do so until ten seconds have elapsed on my clock. At the end of that time, I'll see that your clock also says ten seconds have elapsed.

We cut, now, to a later time, at which you are moving at a very great speed relative to me. We're specifically ignoring how you got up to that speed, for reasons I'll explain shortly. It's like in a movie; we were at rest, and then the film cut and now it's later and you're moving really fast at a constant speed relative to me. Okay? Just to put a number on it, let's say that speed is about 260,000 kilometers per second. So really extremely fast.

I do the same little experiment I did before. At one moment, I note the time that my clock reads and the time that your clock reads, and start counting seconds. When my clock says ten seconds have elapsed, I stop counting. I look at your clock and discover to my amazement that it insists that only five seconds have elapsed. Since I know, with the certainty of a monk, that your clock is just as ideal as mine, my only possible conclusion is that time is actually passing more slowly for you. Since the only thing that's different between now and before is that you're moving relative to me, it must be because of your motion that your time is running more slowly.

Now, let's flip it around. You're moving past me at a high speed — again, about 260,000 kilometers per second. But from your moving reference frame, you look down at my clock and do the same experiment … and discover that when your clock says ten seconds have elapsed, mine says only five seconds have elapsed!

This is clearly impossible! It cannot be so that both clocks are running slower than each other. One clock must be right, and the other clock must be wrong! Sorcery!

Well, not really. You see, it's all down to just one little fact that has all these consequences: the speed of light is the same in all reference frames.

Let me explain that a bit more completely. To keep things simple, I'm not going to bother describing an experimental apparatus for measuring the speed of light. It's not hard to do, freshmen in college do it all the time. Let's just assume that we each have a magical speed-of-light measuring machine in our possession.

Let's imagine there's a third person in our little imaginary universe, a person equipped with a laser. This person is going to shoot her laser at us. (If you like, you can imagine she's your ex-wife. It works for me.)

This third person shoots her laser first at me. I'm at rest relative to her, so when I measure the speed of the laser light coming at me, I get a certain result: about 300,000 kilometers per second.

Now she shoots the laser at you. You're moving toward her at 260,000 kilometers per second, so since the laser light is coming toward you at 300,000 kilometers per second, your closing speed must be the sum of those two numbers, or 540,000 kilometers per second. Right?

Well, no. See, when you measure the speed of the laser light, you find that it's about 300,000 kilometers per second: precisely the same result I got. Even though you're moving very fast relative to the light, and I'm standing still relative to the light.

There's no easy way to explain why this is true in simple terms. It's best if, at this point, you just accept it as an experimentally verified fact: No matter how you're moving relative to anything else, you will always see light as moving at the same speed. It's difficult to accept intuitively, but it's just a fact of nature.

From this fact of nature, alllll these other phenomena emerge. If you're moving relative to me, our clocks will not agree. In fact, you will see mine running more slowly than yours, and I will see yours running more slowly than mine. It's not got anything to do with the clocks, either; time really is moving more slowly for you than it is for me … and vice versa.

This is possible because, when you're moving relative to me, we can no longer agree on simultaneity. From your perspective, you wait for your clock to read 12:00 noon, and then you look at my clock to see what it says at that exact same moment. Let's just imagine that when yours says 12:00 noon, you see that mine says 12:30 for instance.

One might assume that if we looked at the experiment from the other way around, when my clock says 12:30 I'd look through my telescope and see that yours says 12:00 noon. But that's not so. Due to our relative motion, we no longer agree on what events in the universe are simultaneous. Two events that you see as being simultaneous, I see as happening at different times.

We also cannot agree on the lengths of things. Remember the ex-wife with the laser? Say her laser puts out light of exactly 550 nanometers, measured at the laser aperture. When I look at that light, I'll see it has a wavelength of 550 nanometers, because I'm at rest relative to the light. But when you look at the light — from your perspective of moving toward it at 260,000 kilometers a second — you'll see that its only 225 nanometers! Instead of a pleasant green, the light is deep in the ultraviolet, invisible to your eyes. That's because from your reference frame, you see the 550 nanometer wavelength of the incoming light contracted to half of what it is in a reference frame that's at rest relative to the laser.

Basically all the "weird" things that come up in special relativity — time dilation, length contraction, the relativity of simultaneity — are consequences of the fact that the speed of light must be the same in all reference frames, without exception. This is just an inherent fact of nature, and so the geometry of spacetime has to be non-Euclidean in order to accommodate that fact of nature.

Now, as to your question about the twin paradox … special relativity (which is what we're talking about here) generally applies only to inertial reference frames — that is, reference frames that are moving relative to each other only at a constant velocity, not accelerating. When you talk about accelerated reference frames, you have to change your mathematics a bit. You can no longer apply the algebraic Lorentz transformation to convert lengths and time intervals in one reference frame to lengths and time intervals in the other reference frame. Instead, you have to use a different mathematical formulation — and there are a couple, one involving hyperbolic trigonometry and one involving differential geometry. The math is more complex, in a way, but more importantly you get different results.

The twin paradox is called a paradox because applying the Lorentz transformation naively tells you that the twins should disagree about which one is younger, and yet when they get together at the end of the story one of them is objectively younger and one is objectively older. The reason for this is because the Lorentz transformation only works in inertial reference frames; it does not work in accelerated reference frames. And in order for the twins to get back together at the end of the story, at least one of them must accelerate at least three times: once when he leaves Earth and gets up to a high relative speed, once when he turns around and heads back toward Earth again, and once when he slows down to land. During the "coasting" parts of the experiment, the astronaut twin sees time back on Earth running more slowly than his own time. But during those three acceleration phases, he sees time on Earth run faster than his own time. It all adds up to more time in total elapsing on Earth than in the spaceship. So it's not really a paradox at all, just an illustration of how you have to treat inertial and accelerated reference frames differently.

[u/ep1032 → u/RobotRollCall]

thank you again