# Why force due to gravity or gravitation cannot be measured.

1. Dec 16, 2011

### rvn

Its interesting that eventhough gravitation is a force, it cannot be measured. Literatures says that only gravitational gradient can be measured from gravity tensor is to be computed and by applying mathemaatical operations, derive the gravitational vector.
Can anyone help me to understand that why gravitation or force due to gravity cannot be measured.
can accelerometers serve this purpose?

2. Dec 16, 2011

### D H

Staff Emeritus
Accelerometers cannot measure gravitational acceleration.

The classical (Newtonian) explanation is that accelerometers measure acceleration due to all real forces except for gravitation. The general relativistic explanation is that accelerometers measure acceleration due to all real forces (period). Gravitation is a fictitious force in general relativity.

Accelerometers ultimately depend on some test mass that is not rigidly connected to the accelerometer case. The small size of an accelerometer means that the gravitational acceleration of the accelerometer case versus that of the test mass are essentially the same. The only way to make an accelerometer that could sense gravitational acceleration would require somehow shielding the test mass from gravitation. As there is no such thing as a gravity shield, there is no such thing as an accelerometer that can sense gravitational acceleration.

3. Dec 16, 2011

### olivermsun

I don't quite understand the above explanation. Suppose you have the familiar arrangement of an ideal spring-mass type "accelerometer." The mass has a rest position when the the system is oriented sideways. When you dangle it one way, the mass has a new rest position, corresponding to a balance between the spring force and the gravitational force, and when you dangle it the other way, the other thing happens, etc., etc.

Now arguably the mass isn't accelerating, so all you're measuring is the gravitational force and not the acceleration, but it seems fair to infer g by using F = mg. Isn't this about as direct a measurement as you get from any instrument? Or the equivalence of gravitational mass and inertial mass what is being objected to here?

What am I missing?

4. Dec 16, 2011

### D H

Staff Emeritus
Suppose you are in an enclosed room with no windows. All you have is an accelerometer, and it registers an acceleration of 9.80665 m/s2 upwards. How can you tell whether you're enclosed room is sitting motionless on the surface of the Earth or is in a spacecraft out in deep space accelerating at 9.80665 m/s2?

The answer is that you can't tell the difference per Einstein's equivalence principle. Einstein called this little room an elevator car. See http://www.einstein-online.info/spotlights/equivalence_principle.

The accelerometer is not measuring gravitational acceleration. It is instead measuring acceleration with respect to a local inertial frame. There's a significant difference between inertial frames in Newtonian mechanics and inertial frames in general relativity. Inertial frames in general relativity are free-falling frames.

5. Dec 16, 2011

### olivermsun

My instinctual response (not meant facetiously at all) as an observationalist is to avoid calling what I've measured "gravity" unless I can check by opening a window. I mean, you can "measure" centrifugal force with an accelerometer too, even though it too is fictitious.

Last edited: Dec 16, 2011
6. Dec 16, 2011

### D H

Staff Emeritus
Accelerometers do not sense the fictitious centrifugal force. It's a fictitious force. Just like gravity.

Another way to look at it: Suppose you attach accelerometers to your head and your feet and hopped into a roller coaster. Those accelerometer readings are going to vary wildly over the course of the ride. Yet gravitational acceleration has only changed by a tiny, tiny amount. When the roller coaster goes through a zero g roll, the readings from the accelerometer attached to your head will point in a very different direction from that indicated by those attached to your feet. Those accelerometers are not measuring gravitational acceleration. They are measuring acceleration due to all real forces except gravitation.

Another way to look at it: An accelerometer at rest on the surface of the Earth will indicate that the accelerometer is accelerating at about 9.81 m/s2 upward. Gravitational acceleration is of course downward. So something is very wrong here if accelerometers sense gravitational acceleration.

Yet another way to look at it: An accelerometer attached to the International Space Station will register near zero acceleration. Yet the gravitational acceleration on the space station is reduced by only about 10%. Something is very, very wrong here if accelerometers sense gravitational acceleration.

One final way to look at it: In Newtonian mechanics, all observers, inertial or non-inertial, will agree on the real (non-fictitious) non-gravitational forces acting on the accelerometer. Sum these non-fictitious, non-gravitational forces, divide by mass and you will get the accelerometer reading. It doesn't matter if the accelerometer is attached to a skydiver, a spaceship, a roller coaster, or the ground. The simplest explanation in terms of Newtonian mechanics is that accelerometers sense acceleration to all real forces except for gravitational acceleration.

The explanation from the perspective of general relativity is even easier: Accelerometers sense acceleration due to real forces. Period. No need to say "except for gravitational acceleration" because gravitational acceleration, like centrifugal acceleration, is a fictitious force.

7. Dec 16, 2011

### olivermsun

I understand your examples. The sticking point seems to be about what has been "measured." If I say the accelerometer has measured the centrifugal force then a response could very well be to say, no, what has been measured is the inward acceleration which causes one to perceive the (fictitious) centrifugal force. In reality, what has been measured is the tension on a spring which holds pieces of the accelerometer together, and what you object to is calling this a "measurement" of local gravitational acceleration.

Last edited: Dec 16, 2011