Exploring the Effects of Relativity on a Falling Body

[garys_2k]

A "Simple" Relativity Question...

I was asked this question on another forum, and it seems easy at first glance but may be more subtle: What would an observer on a very massive star (yeah, a well insulated observer) see about a small body falling in toward the star, from very far away?

Would the apparent mass of the body change? If so, how would you calculate its mass at any point during the fall? Ditto the incoming body's speed -- what would it be as if fell? All the questions from the reference of the observer on the star.

Obviously SR will influence the apparent mass and speed, but what about GR? Does a freefalling body undergo GR changes? Also, does the fact that the observer is sitting in a pretty deep gravity well influence what he measures about that falling body?

I thought I knew how to handle these cases, but when put all together I'm not sure what the net effects are.

Thanks in advance for any help!

 

[Javier]

In modern language, "mass" is taken to automatically mean the "rest mass" of the object. Therefore, there are no changes to the mass of the object, not in SR, not in GR. The energy an object has can change from one observer to another (but it does so in classical mechanics with kinetic energy as well). Mass is a *part* of the energy the object has.

The spacetime geometry telling the object how to move in this case is not as novel as the geometry as you approach the horizon of a black hole. The trajectory will qualitatively look the same as an object falling toward Earth (of course, we know that as it speeds up, it never reaches light speed...this is a constraint from the spacetime geometry).
What else does relativity say? It says that an observer on the surface of the massive planet will see the passage of time as being slower for the falling object as it speeds up toward the planet. This is the same effect as in the instructive example of infalling muons from "cosmic radiation". They live longer than we'd expect them to due to their relative motion. There are two effects, though: one due to the fact that observers in relative motion have different notions of space and time, the second due to the curvature of spacetime, such that observers at different locations in that curved geometry have different notions about space and time.

 

[R0man]

The effects of relativity on a falling body can be quite complex and are dependent on both special relativity (SR) and general relativity (GR). In this scenario, the observer on the massive star is in a strong gravitational field, which means that both SR and GR must be taken into account.

First, let's consider the apparent mass of the falling body. According to SR, the apparent mass of an object increases as its speed approaches the speed of light. This means that as the body falls towards the massive star, its apparent mass will increase due to its increasing speed. However, in GR, the strong gravitational field of the star will also affect the apparent mass of the body. This is known as gravitational time dilation, which causes time to pass slower in a strong gravitational field. As a result, the falling body will appear to have a lower mass to the observer on the star, due to the slower passage of time in the strong gravitational field.

Next, let's consider the speed of the falling body. In SR, the speed of an object is relative to the observer's frame of reference. This means that the observer on the star will measure the speed of the falling body to be less than the speed of light, even though the body may be accelerating towards the star at a high rate. In GR, the strong gravitational field of the star will also affect the speed of the falling body. Due to the curvature of space-time caused by the massive star, the falling body will experience a change in its trajectory, causing it to accelerate towards the star at a slower rate than it would in a weaker gravitational field.

Finally, we must consider the effects of the observer's own gravitational field on the measurements. As the observer is in a deep gravitational well, their own measurements will be affected by gravitational time dilation. This means that they will perceive the falling body to have a lower mass and slower speed than an observer in a weaker gravitational field would.

In summary, the observer on the massive star would see the falling body as having a higher apparent mass, but a slower speed due to the effects of both SR and GR. The observer's own gravitational field will also affect their measurements, causing them to perceive the falling body as having a lower mass and slower speed than an observer in a weaker gravitational field would. Overall, the net effects of relativity on a falling body in this scenario would be a combination of both SR and GR.