^{1}

^{3}.

By taking the massless limit of linearized massive conformal gravity coupled to a source, we show that the theory is free from the vDVZ discontinuity. This result is confirmed when we take the massless limit of the gravitational potential of the theory, which is shown to be finite at the origin.

Recently, it has been explicitly shown that the classical theory of massive conformal gravity (MCG) has a negative energy massive tensor field [

At the quantum level, the corresponding quantum state of a negative energy field is taken to have negative energy or negative norm [

In quantum MCG, the negative energy massive tensor field gives rise to a massive spin-

The aim of this work is to verify part of the consistency of classical MCG. Due to the extent and complexity of the calculations on the stability of the classical MCG solutions, we will leave them for future works. Here we focus on studying the continuity of the theory in the massless limit. In Section

Let us consider the total MCG action (here we consider units in which

It is worth noticing that under the transformations

Since the integral of the Euler density

Taking the trace of (

Using the weak-field approximations

In order to obtain a second-order derivative form, we choose the method of the decomposition into oscillator variables [

In order to reveal the complete particle content of (

Taking into account the conservation and the traceless condition of

In the massless limit

The gauge conditions (

It is worth noticing that MCG differs from theories of gravity with only a massive tensor field, for which general relativity (GR) should be recovered in the massless limit. Usually it is the absence of this recovery that is associated with the vDVZ discontinuity. However, the real meaning of the vDVZ discontinuity is that the massless limit of a massive theory is not equivalent to the massless theory, which violates the physical continuity principle [

Recently, it was proposed that the vDVZ discontinuity of a local gravitational theory is closely related to the finiteness of the gravitational potential at the origin [

The variation of (

In the case of a point particle source with mass

We have found that MCG contains only two linearized fields with physical degrees of freedom: the usual positive energy massless tensor field and a negative energy massive tensor field. In the massless limit, the number of physical degrees of freedom of the theory remains the same, with the massive tensor field being decomposed into massless tensor, vector, and scalar fields. The absence of coupling between the vector and scalar fields and the source makes the theory free of the vDVZ discontinuity. In addition, it was shown that the MCG potential is singularity free, which indicates that the vDVZ discontinuity is not directly related to the finiteness of the gravitational potential at the origin.

As we saw in Section

The variation of (

With the help of the conformal transformations

It is well known that the weak-field limit of (

No data were used to support this study.

The author declares no conflicts of interest.