Acceleration and Gravity Equivalent?

Suppose a tall (100m high) rocket sits on the launch pad. It is equipped with launch boosters and a sustainer motor that can give the rocket a prolonged 1g length-wise acceleration in free space. Amongst others, it is also fitted with the following sensors: two identical, synchronized atomic clocks, one in the nose and one near the tail, plus three accurate identical accelerometers, one in the nose, one in the tail and one midway between the first two.

While on the launch pad (waiting for a long delayed launch) you monitor all the sensors and determine that the nose clock is marginally gaining time on the tail clock. You satisfy yourself that this is normal due to gravitational time dilation and amounts to 1 part in about 10^14 (coming from dt/t = gL/c^2, where g = 9.81 m/s^2 and L is the difference in height between the clocks). You also verify that the higher accelerometers read marginally lower accelerations than the lower ones, in agreement with the inverse square law of gravitational acceleration (a = -GM/r^2, where G is Newton’s gravitational constant M the mass of the Earth and r the distance from Earth’s center).

Eventually the system is launched into free space and all the boosters fall away. After verifying that everything operates as designed and synchronizing the nose- and tail clocks, you ignite the 1g-propulsion system at the back. After a fair time of monitoring exactly 1g of acceleration at the tail of the rocket, you read all your sensors again. Will the clocks and accelerometers be able to tell you that you are now being linearly accelerated at 1g in free space and no longer sitting stationary on Earth’s surface?

67 COMMENTS

  1. I have recently learned that all the effects inside an accelerating laboratory are exactly equivalent to the gravitational effects of an ‘infinitely long’ cylindrical mass (at least in two of the three spatial dimensions).

    If the lab is kept statically at a radial distance r perpendicular to the cylinder’s long axis, the gravitational acceleration scales with 1/r, just as is the case in a tall accelerating lab.

    In the transverse direction parallel to that axis, there is no tidal forces focusing to the ‘center’ of the cylinder (an infinitely long thing has no defined center), exactly as inside the accelerating lab.

    So purely inside such a lab, there will be no way to distinguish between gravity and linear, uniform acceleration, unless you use the third dimension, perpendicular to the both the radial and the axis. In that direction, there will be the squeezing effect (of tidal gravity) to the centerline of the cylinder.

    Interesting.

    Burt Jordaan (www.Relativity-4-Engineers.com)

  2. I agree that the Principle of Equivalence has most widely come to be applied to that of inertial and gravitational mass. But originally it was more ambitious, it was Einstein’s seminal concept that led to the General Theory. I don’t have it at-hand, but he stated explicitly that he wanted to demonstrate the relativity of all accelerating frames of reference, just as he had uniform frames – and accordingly, gravitation in one frame of reference was held to be equivalent to inertial acceleration from another. That was the original “general” in the General Theory.

    The original principle, that all forms of acceleration are equivalent, has morphed into various more moderate interpretations. But even that of inertial and gravitational mass is open to criticism. Given that the weight we experience at the earth’s surface is the resistance to our geodesic motion, unless we are sinking in water or quicksand, it is exactly equal to the force needed to offset the geodesic motion of our inertial mass. Gravitational mass is equivalent to inertial mass because gravitational mass IS inertial mass.

  3. The fact that Jim claims to be able to distinguish between gravity and an accelerated frame in local measurements means that he can distinguish between gravitational and inertial mass.

    At least that’s my reading of his statements, especially as regards the (ir)relevance of Einstein’s thought experiment that leads to that equivalence.

    Now onward to Scruffy’s new thread.

    Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

  4. The fact that Jim claims to be able to distinguish between gravity and an accelerated frame in local measurements means that he can distinguish between gravitational and inertial mass.

    At least that’s my reading of his statements, especially as regards the (ir)relevance of Einstein’s thought experiment that leads to that equivalence.

    Now onward to Scruffy’s new thread.

    Fred Bortz — Science and technology books for young readers (www.fredbortz.com) and Science book reviews (www.scienceshelf.com)

  5. Hi Jim, also consider my last reply to Fred.

    Not sure one should read too much into the “principle of equivalence” between gravity and acceleration. It’s rather about the “principle of equivalence” between gravitational- and inertial mass.

    Burt Jordaan (www.Relativity-4-Engineers.com)

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