Inertial Reference Frames: Study with Video Lessons, Practice Problems & Examples

Inertial reference frames are crucial in special relativity, defined as frames moving at constant velocity. They can be categorized into rest frames, like a lab on Earth, and moving frames, such as a car in motion. Non-inertial frames, which involve acceleration, are generally ignored in special relativity. Understanding these concepts is essential for analyzing events, such as the decay of particles or the ticking of clocks, as measurements differ between frames. The proper frame moves with the event, emphasizing the relativity of motion and the absence of absolute rest or motion.

Inertial Reference Frames

Hey guys. In this video, we're going to talk about one of the most important concepts in special relativity, which is the concept of an inertial reference frame. Okay? Let's get to it. Now, you can broadly break up reference frames, which are just coordinate systems that you construct arbitrarily to measure things. Right? If we're trying to measure displacement, let's say, if an object starts at some point x, y, and moves to some new point x₁, y₁, the only way that we can say that it started at some position, x, y, and ended at some new position is if we construct a coordinate system, right, in this case, a Cartesian x y regular horizontal vertical coordinate system, to measure the initial and final position. So that's coordinate system dependent, and the coordinate system just represents a reference frame. Something with which we can refer to, right, to make a measurement. So you can broadly break up reference frames into 2 categories: inertial reference frames and non-inertial reference frames. Okay? Inertial reference frames are ones that move at a constant velocity. Okay? So the frame itself is either at rest or moving at a constant velocity. A perfectly good example is a frame which is at rest. Right? Just you standing somewhere in a lab making measurements. A good example of a moving inertial frame is a car, for instance. If your car is going at 20 miles an hour, when you're sitting in the car you feel like you're at rest and you can make measurements inside the car within your reference frame. But to an observer outside the car standing on the side of the road, they see your reference frame moving at a constant velocity. So the measurements that you make in your reference frame aren't going to be the same necessarily as the measurements the outside observer makes in his reference frame. But they're still inertial frames because your car is moving at a constant velocity and the guy on the sidewalk is at rest. Both of those are inertial frames. Okay. Non-inertial frames, as the name implies, are frames that are not inertial. And if inertial frames move at a constant velocity, non-inertial frames move at a changing velocity, which means, right, that they have acceleration. Okay. So accelerated reference frames are non-inertial frames. Okay? Constant velocity frames are inertial frames. Okay? So, inertial frames are typically subdivided into 2, once again, broad categories, the lab frame and the moving frame. Okay? Or sorry. Rest frames and moving frames. Rest frames are frames that have a, quote-unquote, 0 velocity. And I put the quotes there specifically, and I'll get to those in a second. Moving frames, as the name implies, have a quote-unquote non-zero, right, velocity. The lab frame is the most common type of rest frame, and it's just a frame that's at rest with respect to the earth or specifically the earth's surface. Okay. That's where you construct a laboratory. You build it on the surface of the Earth. You put tables. You put instruments, and you're making your little measurements inside that stationary reference frame in the lab. Okay. That is by far the most common type or the most common example of a rest frame. Frames that move at some velocity. Okay? Sorry. At the same velocity as some event is specifically how I worded it here. And this is typically how special relativity is done. You're interested in events. You're interested in things that are happening. Okay? So if that event, let's say it's a particle that's unstable that can decay. Okay? So there's a little atom that's moving along. It's not very stable so eventually it's going to break apart. That decay, that breaking apart, that's the event that you're interested in. That atom can be moving very quickly in the lab frame. Like 1% the speed of light, 10% the speed of light. The lab frame is what the scientists at rest in the lab and they're going to see that atom whizzing by. But if you were to construct a reference frame that moves with the atom, that would be considered what we would call the proper frame. Okay. The proper frame is the one that moves at the same velocity as an event. Right? I put event in quotes. That's exactly what I was just explaining, that we care a lot about events when discussing relativity. And so the proper frame is the one that's going to be going with the event. One of the most common events that you're going to discuss in relativity is going to be a ticking clock. I know that doesn't really sound like an event, but this is typically how physicists phrase it. A ticking clock can be moving or can be stationary. For instance, you could be standing, and on your wrist you could have a little watch, right, that's ticking by. Or somebody who's in a car moving past you could be wearing a watch that's ticking by. If you are interested in this guy's watch, then the moving frame, the frame moving with the car is the proper frame. Okay? Now something that I need to key in on because I said it specifically here are these quotes. Okay? Why are they in quotes? Why is it 0, and nonzero in quotes? Well, it's because that doesn't actually mean anything. What does 0 velocity mean? I said the lab frame, which is the most common type of rest frame, is at rest with respect to the earth. There's no such thing as absolutely at rest or absolutely moving. That's not an actual physical concept. Everything is moving relative to one another or at rest relative to one another. But there's no sort of universal coordinate system where you can say something is definitely at rest or something is definitely moving. Typically, special relativity problems are going to sort of be anchored to the earth. Okay? Because that gives us as people. Right? I mentioned it here, us as humans. Right? Us as people, an easier way to understand the problems and what's going on. If you anchor things to the surface of the earth, you see the surface of the earth is at rest. If you're measuring things relative to the surface of the earth, they're at rest. If you're measuring things moving relative to the surface of the earth, then they're in motion. That is conceptually a lot easier for people to understand. Out in space, though, which there are quite a bit of special relativity problems that are not anchored to the earth but occur out in space, you can't use the Earth as a reference point. You can't say that the Earth is stationary. And so everything's stationary relative to the earth is stationary; everything moving relative to the earth is moving. You can't say that because you don't have that anymore. So you just have to arbitrarily choose one lab frame. So a popular instance or a popular example is, let's say, that there is a spaceship chasing another spaceship. You can consider both of these to be moving frames if you want, or you can consider one of them to be stationary, and then the other is moving relative to the stationary one. Okay? This is all stuff that's probably really confusing right now, and it's really confusing to everybody when you first see it. But stick with it, and what you'll see as we start covering problems is that it'll start to make more and more sense. It'll start to click. Okay? But these reference frames are really hard to get used to, at first. Okay? So near the surface of the earth, right, we would typically consider, right, a lab frame to be at rest relative relative to the surface of the earth, and a moving frame to be moving relative to the center of the earth. We usually call lab frames as s and the moving frames as s prime. Now that's entirely up to your professor, your book, what is what. But that's just the typical convention that I've always come across, and that's the convention that we're going to use in these videos. Okay? And also, u is typically used for the velocity of a frame, whereas v will be used for velocities of things within the frame. Okay? So if you see this u right here is the velocity of that s prime frame, the moving frame, relative to the surface of the Earth. And conversely relative to the lab frame over there, frame s. Right? Because s is at rest with respect to the Earth, so if s prime is moving at u with respect to the earth, it's also moving at u with respect to the lab frame. Okay? Because the lab frame, once again, is at rest with respect to the earth. So if there is some object moving at a speed v, or velocity v, I could say, in the lab frame, if we were to measure the velocity in the moving frame, it would be a different velocity v prime. Okay? By the way, don't think about these two reference frames as being spatially separated. Okay? I just have to show them separated so that you can pictorially understand it. So you can visualize it. But imagine if this guy was 1 meter into that frame, this guy could be 1 meter into this frame as well. Okay? So this is the difference between s, the lab frame, and s prime, the moving frame. And those velocities v and v_prime, they're not going to be the same. Okay? Now the last thing to discuss before we wrap up this video on inertial frames is that non-inertial frames aren't important. What is important to understand is that they are ignored in special relativity. Okay? Special relativity never deals with non-inertial frames. Special relativity only deals with inertial frames. Okay? General relativity, the second theory of relativity published by Einstein much later on, actually 10 years later, that deals with non-inertial frames. Okay? Now, technically, the earth, the surface of the earth, is moving in a circle. And since it's technically moving in a circle, it is actually a non-inertial reference frame. Right? It's just that the earth rotates very very slowly, so we don't really notice the rotation of the earth. When you're in a car and the car starts to accelerate, you feel the acceleration of the car. But when you're just standing on the surface of the earth, you do not feel that acceleration because it's almost unnoticeable at such a small acceleration. Okay? Now the fact though that the earth is a non-inertial reference frame does actually have real-life ramifications. Specifically, there's the Coriolis force, which is responsible for how hurricanes rotate, and then the centrifugal force, which actually alters slightly the gravitational acceleration at the pole of the earth and at the equator. So if you have the earth and here's the equator, the gravitational acceleration here is going to be slightly different than the gravitational acceleration would be at the equator because of the centrifugal force. It's an almost it's a very very very small and hard to measure difference, but the difference is actually there. Okay? But once again, these aren't important to our discussions. They're just real-life ramifications of non-inertial frames. Alright, guys? So that wraps up this introduction to inertial frames. Even if you don't quite understand them at this point, that's okay because we're going to be using them continuously throughout our discussion of special relativity. And the best way to really understand them is to start seeing problems where we start using those inertial frames. Alright? Thanks so much for watching guys, and I'll see you in another video.

Here’s what students ask on this topic:

What is an inertial reference frame in special relativity?

An inertial reference frame in special relativity is a coordinate system where an object either remains at rest or moves at a constant velocity. This means there is no acceleration in such frames. These frames are crucial for analyzing physical phenomena because the laws of physics, particularly those of special relativity, hold true in these frames. Examples include a stationary lab on Earth or a car moving at a constant speed. Inertial frames are contrasted with non-inertial frames, which involve acceleration and are not considered in special relativity.

How do measurements differ between inertial reference frames?

Measurements differ between inertial reference frames due to relative motion. For instance, if you are in a car moving at a constant velocity, you might measure the speed of an object differently than someone standing on the sidewalk. This difference arises because each observer is in a different inertial frame. Special relativity provides the tools to transform measurements from one inertial frame to another, ensuring that the laws of physics remain consistent across all inertial frames.

What is the difference between inertial and non-inertial reference frames?

Inertial reference frames move at a constant velocity, meaning they do not experience acceleration. Examples include a stationary lab or a car moving at a constant speed. Non-inertial reference frames, on the other hand, involve acceleration. This could be a car that is speeding up or slowing down. Special relativity focuses on inertial frames, while general relativity deals with non-inertial frames. Non-inertial frames can introduce fictitious forces, such as the Coriolis force, which are not present in inertial frames.

Why are non-inertial frames ignored in special relativity?

Non-inertial frames are ignored in special relativity because the theory is built on the principle that the laws of physics are the same in all inertial frames. Special relativity deals with objects moving at constant velocities and does not account for acceleration. Non-inertial frames, which involve acceleration, introduce complexities that are addressed by general relativity. General relativity extends the principles of special relativity to include non-inertial frames and gravitational effects.

What is a proper frame in the context of special relativity?

A proper frame in special relativity is a reference frame that moves with the event being observed. For example, if you are studying a particle that decays, the proper frame would be the one moving with the particle. This frame is crucial for accurately describing the event because it simplifies the analysis by eliminating relative motion between the observer and the event. Proper frames are often used to measure time intervals and distances in the context of special relativity.