MadSci Network: Physics

Re: If entire universe is accelerating how can Sp. Relativity apply?

Date: Fri May 16 09:22:15 2008
Posted By: Jim Guinn, Staff, Science, Georgia Perimeter College
Area of science: Physics
ID: 1210359712.Ph

Dear Ray,

This is a great question and it really emphasizes the need to understand 
two different kinds of acceleration, and the addition of one more 
relativity postulate called the Equivalence Principle.

It is true that the first postulate of Special Relativity states basically 
that all the laws of physics are the same in all inertial reference 
frames, and an inertial reference frame is a coordinate system that is not 
accelerating.  (The second postulate being the universality of the speed 
of light.)  This typically means we need to restrict Special Relativity to 
frames which are moving at a constant velocity and are not near any strong 
gravitational fields.  This avoids any accelerations caused by any forces, 
gravitational or otherwise.

To make the jump to General Relativity we need to include another 
assumption in addition to the two for Special Relativity.  This third 
postulate is the Equivalence Principle.  The Equivalence Principle is 
sometimes broken down into two parts, the Weak and Strong Equivalence 
Principles, but the basic idea is that the force of gravity is equivalent 
to an acceleration of the reference frame.  To give an example, imagine 
two spacecrafts, one resting on the surface of the Earth, where there is a 
gravitational field causing everything to fall down with an acceleration 
of 9.8 m/s/s , and the other spacecraft out in space, away from any 
gravity, but which is accelerating through space with an acceleration of 
9.8m/s/s .  In both crafts, a “dropped” object would approach the back of 
the ship with an acceleration of 9.8 m/s/s .  In the ship on Earth, the 
object falls toward the ground due to gravity, but in the ship moving 
through space, the back of the ship would move forward toward the object.  
The important point is that for someone in the ship, there is nothing that 
can be done to distinguish between these two situations.  So, the effects 
of gravity are exactly the same as the effects of the frame accelerating.  
Why is gravity so important in this example?  Why couldn’t some other 
force be acting on the “stationary” ship and be equivalent to an 
accelerating frame?  Well, it seems that only gravity acts “equally” on 
everything, that is, with a magnitude proportional to the object’s mass, 
or measurement of its inertia.  All other forces depend on some other 
property, e.g. the electrostatic force depends on the object’s charge.  
So, if a space ship was sitting in an electric field, only charged 
particles would experience a force, and it would be obvious that the whole 
ship was not accelerating.

So, are there any inertial reference frames in a gravitational field, 
since gravity is like an accelerating frame, and accelerating frames are 
not supposed to be inertial?  As a matter of fact, there are!  A frame can 
actually remove the effects of gravity if it is allowed to free fall.  
That is, if the frame is allowed to move freely under the influence of 
gravity.  The gravitational field will act on everything in that frame 
giving it the same acceleration (if we assume the field is uniform within 
our area of interest).  To someone in that frame, it would appear that the 
frame was out in space, away from any gravity and not accelerating!  (At 
least until the frame hit the ground!)  We usually refer to this 
as “weightlessness”.  Therefore, according to General Relativity, we still 
have inertial frames, but they must be accelerating if they are in a 
gravitational field.  This is a very different idea than in Special 
Relativity where the frames could not be accelerating.  Therefore, from 
General Relativity, we now see that a freefalling frame is inertial, and 
therefore all the laws of physics hold in a freefalling frame.  One set of 
the laws of physics being Special Relativity!

This must seem like a roundabout way to finally get to your question, but 
now we are ready to understand it.  Is the Earth accelerating through 
space?  Yes, it is in orbit around the Sun.  Does Special Relativity hold 
on the Earth?  Yes, it does!  Where does it hold if the Earth is 
accelerating?  Since the Earth is in free fall around the Sun … Hold it 
right there!  The Earth is in free fall around the Sun?  Well, yes, since 
the Earth is moving in its orbit under the influence of the gravitational 
field of the Sun, it is in free fall, actually, that’s what all orbits 
are.  So, for the most part, we don’t notice the gravitational field of 
the Sun here on the Earth, since we are constantly moving under its 
influence.  (This is not completely true because the Sun’s gravity is not 
uniform, and so one side of the Earth feels slightly more pull than the 
other, and along with the Moon’s gravity [mostly the Moon’s gravity] 
causes our tides.)  The acceleration of the Earth around the Sun does not 
cause the break down of Special Relativity.  What about the gravitational 
field of the Earth?  Well, yes it could.  If an experiment were done in a 
freely falling frame on the Earth, then Special Relativity would 
absolutely hold.  In a frame bound to the Earth, General Relativity 
certainly may cause some changes to a Special Relativistic prediction, 
however, since on the Earth, gravity acts “vertically”, Special Relativity 
is still very accurate for “horizontal” experiments.

Well, Ray, I hope that answers your question.  You are right in that in 
general all matter is accelerating due to gravitational fields, but that 
does not preclude using Special Relativity.  If we can do an experiment in 
a frame which is also accelerating due to gravity, then that frame is 
inertial, and therefore Special Relativity still holds!

Thank you for your interest and please let us know if you have any other 


Jim Guinn
Georgia Perimeter College

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