Dr. Aaron Titus | Department of Physics, High Point University PHY1050      Astronomy of Stars, Galaxies, and the Cosmos home | WebAssign | textbook | course calendar

## inertial reference frames

Special relativity only applies to inertial reference frames. An inertial reference frame is a frame in which Newton's first law of motion is valid. That is, if there are no forces acting on an object, it should remain at rest or move with a constant speed in a constant direction.

One way that you can test whether your reference frame is inertial or not is to do a two-ball-test. (No, this is not some great definition or something. It's just a typical Titus "silly name"). The two-ball-test is this: if you release two balls from rest and they remain at rest, then you are in an inertial reference frame (at least during the time that you are watching them).

To illustrate this, check out the simulation below. Suppose that you are sitting in a bus that is at rest, and you set two balls down on the floor in the aisle. If the bus is at rest, what will be the motino of the balls? (It's an obvious answer, I know.) Click the link for Bus at rest to view the motion of the balls. (You will have to click the Play button to start the simulation.)

Obviously the balls remain at rest. The laws of physics are valid. This is an inertial reference frame.

Now, suppose the bus speeds up and moves to the right? Think about what the balls will do. Now, click the link for Bus moving to the right and speeding up to see the motion of the balls. They start out at rest and then accelerate to the left. Yet, there are no thrusters, nobody pushing on the balls, or another force to make them accelerate. They appear to accelerate though there are no forces on the balls! That violates our laws of physics! Therefore, the bus in this case is NOT an inertial reference frame.

You should be able to use the two-ball-test to say whether a reference frame is inertial or not.

What about something falling near Earth, like an apple falling from a tree? Well, it speeds up as it falls. According to Einstein, the speeding up is not due to some mysterious force called "gravity." But rather, it's an artifact of our reference frame.

Now, think about that statement for a moment. It's just like the bus and balls. Instead of saying, "Oh, there's a force making the balls accelerate backwards," we instead say, "The apparent acceleration of the balls is a result of viewing them in the reference frame of the accelerating bus. There is really no force acting on the balls to make them accelerate."

So, according to Einstein, we are viewing the balls from a non-inertial reference frame. If we change our reference frame, then Newton's first law would make sense again, and a ball released from rest will remain at rest.

Let's visualize this. Suppose that a ball is thrown from a cliff. You view the motion of the ball from another cliff. Click the link, "You are at rest while viewing a projectile." to view the motion of the ball. Be sure to click the Play button to start the animation.

As you might guess, the ball travels along a curved path. Isaac Newton would say that the curved path is a result of the gravitational force of Earth on the ball.

However, Albert Einstein might say, "Rubbish!" (ok, he wouldn't say "Rubbish" but can't I have some artistic license?) "The curved path is a result of our reference frame. If you fall off the cliff as you watch the motion of the ball, then the ball appears to move in a straight line with a constant speed. You are now viewing the ball from an inertial reference frame."

So, according to Einstein, the gravitational force measured near Earth is just a result of our reference frame.

So, check this out. Click the link "You are falling from a cliff while viewing a projectile." Note the motion of the ball. Because you are viewing its motion in an inertial reference frame, it moves at a constant speed in a straight line. You would say that there is no gravitational force.

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