# Accelerated frame vs gravity frame

Can accelerated frame change curvature of space as gravity does? Can there accelerated frame be pure inertial frame?

Can an accelerated frame change curvature of space as gravity does?

No, not essentially. A change of frame simply corresponds to a general coordinate transformation. Curvature is represented by a tensorial quantity called the Riemann curvature tensor. Thus, if it vanishes in one coordinate system (i.e., if the spacetime is not curved in one coordinate system) then it will vanish in all coordinate systems (i.e., an accelerated frame will not see any curvature if the original inertial frame also didn't). However, if the curvature is non-zero then the numerical value of different components of the curvature tensor will change depending on the choice of the coordinate system that is used to describe the curvature--but, geometrically, it would still be the same curvature--just represented in a different choice of the coordinate system. There is an important point to notice here which is not really asked in your question but I would like to point out nonetheless which is that if the curvature is non-zero at some point then it will remain non-zero even in the local inertial frame (mathematically, this can be seen by the virtue of the fact that choosing a local inertial frame only corresponds to making the metric and its first derivatives trivial but the curvature tensor involves second derivatives of the metric which cannot be fully controlled by moving to a local inertial frame).

Can an accelerated frame be pure inertial frame?

No, a frame which is accelerated with respect to a local inertial frame would not be inertial around that spacetime point. The transformations that preserve the property of a coordinate system being locally inertial are fully characterized by three boosts and three rotations and no combination of them creates a frame which is accelerated with respect to the original frame. However, if the curvature tensor vanishes in the original inertial frame, it will continue to vanish in the accelerated frame as well (for the reasons already described) and in that sense, there won't be "true gravity" in the accelerated frame either. But, technically, it won't be an inertial frame in the sense of the metric being Minkowskian and the Christoffel symbols being trivial.

• what is wrong in my argument: Now as inertial frame is defined as the frame in which space is homogeneous,isotropic and time being homogenous and as accelerated frames can't be inertial one's therefore in accelerated frame spacet-time curvature should be changed. – RandomXYZ May 21 at 10:30
• @RandomXYZ in your argument you are saying given “if A then B” and given “not A” you are trying to conclude therefore “not B”. That is a logical fallacy called improper transposition. Here A is “global frame is homogenous and isotropic in space and time” and B is “spacetime is flat”. – Dale May 21 at 12:01
• fallacyfiles.org/imptrans.html – Dale May 21 at 12:04
• @Dale so in conclusion we can say that if frame is non inertial then space can be homogenous and isotropic or not?And also by the same logic can we say that laws of physics can hold true in non inertial or not? – RandomXYZ May 21 at 19:56
• The correct logic would be “if not B then not A” or in other words: if spacetime is not flat then there is no global frame which is isotropic and homogenous in space and time. – Dale May 21 at 21:23