How does weight work in general relativity?

[Ascenxion]

How does http://en.wikipedia.org/wiki/Weight" work in general relativity? Well, since GR states that the gravitational force does not exist...

What about different weights at different altitudes?

 

[mgb_phys]

Exactly the same - you can't unfortunately lose weight by simply believing in GR!

Remember physcial laws are just models.
Newton's laws say - if you pretend there is a force that depends the product of the masses and the inverse square of their distances then objects behave like this. And this is pretty much what we see in experiments, th eforce doesn't have to be real.

GR says - if you imagine that objects moves in straight lines on a curved space then this is how things will behave in nature. It's slightly more correct for a few extreme cases - but in general predicts the same behaviour as Newton's laws.

 

[Ascenxion]

Thanks.
But what about,

What about different weights at different altitudes?

Is it because the space-time curvature (gravitation) decreases slightly by distance?

 

[George Jones]

How does http://en.wikipedia.org/wiki/Weight" work in general relativity? Well, since GR states that the gravitational force does not exist...

What about different weights at different altitudes?

In GR, the "weight" of an object is the magnitude of the object's 4-acceleration multiplied by its rest mass. The magnitude of the 4-acceleration of an object stationary on the Earth's surface is g; the magnitude of the 4-acceleration of an object stationary at an altitude above the Earth equal to the Earth's radius is g/4.

 

[George Jones]

Is it because the space-time curvature (gravitation) decreases slightly by distance?

Not exactly; spacetime curvature is a measure of tidal force.

 

[atyy]

Exactly the same - you can't unfortunately lose weight by simply believing in GR!

In GR, the "weight" of an object is the magnitude of the object's 4-acceleration multiplied by its rest mass.

Two different definitions of weight?

If my weight is my four-acceleration, then assuming I am a test particle, I can lose weight by believing in GR and free-falling

On the other hand, Schutz (A First Course in General Relativity) describes determining the mass parameter of a Schwarzschild object by observation of the trajectory of a test particle at infinity as "weighing" the Schwarzschild object. On this definition, if I am a black hole, I can't lose weight by believing in GR

 

[Ascenxion]

So weight is basically the proper acceleration we experience at surface, according to General Relativity?If so, how does our proper acceleration differ at different points of the Earth?
Thanks.

 

[Naty1]

So weight is basically the proper acceleration we experience at surface, according to General Relativity?


If so, how does our proper acceleration differ at different points of the Earth?

There are two concepts of mass used in physics:mass which resists acceleration (inertial mass) and gravitational mass which describes how mass reacts to a gravitational field. Einsteins equivalence principle says the force from accelerated motion and from a gravitational field are indistinguishable...implying an equivalence between inertial and gravitational mass.

So F= MA=MG=W

to answer your questions, weight is not acceleration...weight is a force.

From W=MG, weight varies by altitude because gravity weakens at greater distances from the center of the earth...you weigh less on a mountaintop than at sea level and even less in an airplane flying over the mountaintop.

By the way, this is of course what Newton says...you don't need relativity.

 

[D H]

In GR, the "weight" of an object is the magnitude of the object's 4-acceleration multiplied by its rest mass.

Two different definitions of weight?

Yep. Two different definitions of "weight". Actually, there are three: Legally, weight is mass. I'll ignore the silly lawyers here and address the different definitions of weight in physics. Classical physics tautologically defines weight ("actual weight") as gravitational force, or mass times acceleration due to gravity. There is one big problem with this definition: It is not measurable. It is, however, a very useful fiction when modeling things such as airplane flight.

Another definition of weight in classical physics is what is called "apparent weight" by some. The apparent weight of some object the net force acting on an object less the gravitational force acting on a body. This latter concept of weight is much more closely aligned with the GR concept of weight than is "actual weight".

If my weight is my four-acceleration, then assuming I am a test particle, I can lose weight by believing in GR and free-falling

We do that in classical mechanics, too. We speak of astronauts in the space station as being "weightless", even though their actual weight while on orbit is 90% of their actual weight on the surface of the Earth. Astronauts in the space station do of course have near zero apparent weight -- and near zero 4-acceleration. (Aerodynamic drag and other perturbative forces makes the apparent weight not quite zero.)

 

[George Jones]

Classical physics tautologically defines weight ("actual weight") as gravitational force, or mass times acceleration due to gravity. There is one big problem with this definition: It is not measurable. It is, however, a very useful fiction when modeling things such as airplane flight

Another definition of weight in classical physics is what is called "apparent weight" by some. The apparent weight of some object the net force acting on an object less the gravitational force acting on a body. This latter concept of weight is much more closely aligned with the GR concept of weight than is "actual weight".

By coincidence, a couple of days before this thread was started, I looked up the definition of weight in three fairly standard first-year textbooks and found ... three different definitions of weight!

1) the force of gravity (a vector) (Walker)
2) the magnitude of the force of gravity (Giancoli)
3) what a scale measures (Knight)

We do that in classical mechanics, too. We speak of astronauts in the space station as being "weightless", even though their actual weight while on orbit is 90% of their actual weight on the surface of the Earth. Astronauts in the space station do of course have near zero apparent weight -- and near zero 4-acceleration. (Aerodynamic drag and other perturbative forces makes the apparent weight not quite zero.)

Even though 1) and 2) are more popular, I tend to favour 3) because then (ideal) astronauts in (ideal) orbit really are weightless, and, as you say, this definition makes sense in GR.

 

[jtbell]

If I remember correctly (don't have the book at hand at home), Paul Hewitt's "Conceptual Physics" which is a very popular textbook for non-mathematical physics courses, also uses definition 3. I like that definition myself, and would prefer to call $mg$ only "gravitational force", but our math-based intro physics courses (both with and without calculus) use definition 2, so I stick with it to avoid confusing our students.