# How can we use Newtons 1st and 2nd Laws on Earth when gravity makes all frames non-inertial?

As Newtons laws of motion only hold in inertial reference frames, how come we use them freely to describe motion of particles under the influence of gravity? Isn't the only frame where we can apply Newtons laws in a frame accelerating at the same rate and in the same direction as gravity?

We can use Newton's Laws in non-inertial reference frames if we consider the action of ficticious forces. Indeed, in General Relativity the frame in free-fall is inertial and the one at rest relative to the ground is non-inertial. But we can use Newton's Laws if we consider the ficticious force: gravity. This is the same thing that happens in the accelerating elevator: we may use Newton's Laws in the elevator frame if we consider the ficticious force field that arises from the equivalence principle.

• Gravity is not a fictitious force and in-fact it is even included in the inertial frame calculations which are normally made. The fictitious forces (or psedo-forces) to consider include coriolis and centrifugal forces due to rotation of earth Feb 28, 2019 at 18:01
• In classical mechanics. Notice he asks: "isn't the only frame where we can apply Newtons laws in a frame accelerating at the same rate and in the same direction as gravity?". He is refering to the General Relativity definition of inertial frame, where gravity is a pseudo force that exists only on the non inertial frame of the groud. Feb 28, 2019 at 18:18
• @sarthak The force of gravity doesn't exist in a true inertial frame. Gravity is precisely the fictitious force that arises in a frame which is accelerated with respect to the local inertial frame. In particular, the Christoffel symbols identically vanish in a local inertial frame and the value of them in a non-inertial frame is determined precisely by the transformation that takes one to the said non-inertial frame from the inertial frame.
– ACat
Mar 1, 2019 at 12:28
• @DvijMankad If we talk about Newtonian physics then force of gravity is a real force which acts between two bodies because of their masses. In GR, there is no force of gravity but just curvature of space-time again due to masses. So gravity is never a fictitious force. A freely falling frame is accelerated at g and the psedo-force appears precisely because the frame is accelerated (in Newtonian terms). But this does not imply that the gravity is a psedo-force. Mar 1, 2019 at 16:57
• @sarthak The quantity that represents itself as a force in GR is Christoffel symbols--not the curvature tensor. Christoffel symbols are always identically zero in an inertial frame and the force of gravity is indeed a fictitious force in GR. It is incorrect to say that there is no force of gravity in GR--there is. It is just that it identically vanishes in an inertial frame.
– ACat
Mar 1, 2019 at 19:24

Inertial reference frames are frames where bodies with zero net force move at constant velocity. It follows from this definition that inertial reference frames have constant velocity with respect to one another. Measurements in one inertial frame can be converted to measurements in another by a Galilean transformation.

If one reference frame is being accelerated you can no longer use a Galilean transformation, and you must include ficticious forces that act on masses whose motion is described using a non-inertial frame of reference.

I think that your confusion comes from the fact that what we usually call "laboratory frame" is in the Earth's surface, so it suffers a gravitational force. However, remember that a normal force is also acting over the objects over the surface of the Earth, so the net acceleration is zero.

• OP is using the GR definiton of inertial frame. Feb 28, 2019 at 17:54
• @João Vítor G. Lima the post is tagged with Newtonian-mechanics Feb 28, 2019 at 17:57
• Because he asked about Newton's Laws right after. Feb 28, 2019 at 18:15
• Notice he asks: "isn't the only frame where we can apply Newtons laws in a frame accelerating at the same rate and in the same direction as gravity?". Feb 28, 2019 at 18:19